The Sarasvati River (Sanskrit: सरस्वती नदी sárasvatī nadī) is one of the chief Rigvedic rivers mentioned in ancient Hindu texts. The Nadistuti hymn in the Rigveda (10.75) mentions the Sarasvati between the Yamuna in the east and the Sutlej in the west, and later Vedic texts like Tandya and Jaiminiya Brahmanas as well as the Mahabharata mention that the Sarasvati dried up in a desert. The goddess Sarasvati was originally a personification of this river, but later developed an independent identity and meaning.
Most scholars agree that at least some of the references to the Sarasvati in the Rigveda refer to the Ghaggar-Hakra River, while the Helmand River is often quoted as the locus of the early Rigvedic river. Whether such a transfer of the name has taken place, either from the Helmand to the Ghaggar-Hakra, or conversely from the Ghaggar-Hakra to the Helmand, is a matter of dispute.
There is also a small present-day Sarasvati River (Sarsuti) that joins the Ghaggar river.
The name Sárasvatī is descended from Proto-Indo-Iranian sáras-wn̥t-iH (virtually PIE *séles-wn̥t-ih2), meaning "she with many pools". Sanskrit saras- means "pool, pond"; the feminine sarasī́ means "stagnant pool, swamp" (e.g. RV 7.103.2b). Cognate with Greek ἕλος "swamp", the Rigvedic term refers mostly to stagnant waters, and Mayrhofer considers unlikely a connection with the root sar- "run, flow".
Sarasvatī is the a Devi feminine of an adjective sarasvant- (which in the masculine occurs in the Rigveda as the name of the keeper of the celestial waters, e.g. 7.96.4, 10.66.5); it is cognate to Avestan *Haraxwaitī, speculated by Lommel (1927) to refer to Arədvī Sūrā Anāhitā, the Avestan mythological world river, which would point to an already Proto-Indo-Iranian myth of a cosmic or mystical *Sáras-vn̥t-iH River. In the younger Avesta, Haraxwaitī is identified with a region described to be rich in rivers, and the Old Persian cognate Hara[h]uvatiš was the name of the Helmand river system, the origin of the Greek name Arachosia. The alteration of the sound "S" to "H" by the Persians is well documented. e.g., the Sanskrit name for Indus is Sindhu, but Persians called is "Hindu", which was later used to describe the river as well as the people living beyond the river. The word Sarasvati may have been adopted in Avestan as Haraxwaitī.
The Sarasvati River is mentioned 72 times in the Rigveda, appearing in all books except for book four.
Sarasvati is mentioned both as the chief of the Sapta Sindhu, the seven major rivers of the early Rigveda, and listed in the geographical list of ten rivers in the Nadistuti sukta of the late Rigveda, and it is the only river with hymns entirely dedicated to it, RV 6.61, 7.95 and 7.96.
The Rigveda describes the Sarasvati as the best of all the rivers (RV 2.41.16-18; also 6.61.8-13; 7.95.2). Rigveda 7.36.6 calls it "the Seventh, Mother of Floods" sárasvatī saptáthī síndhumātā. RV 2.41.16 ámbitame nádītame dévitame sárasvati "best mother, best river, best goddess" expresses the importance and reverence of the Vedic religion for the Sarasvati river, and states that all life spans (āyuṣ) abide on the Sarasvati. Other hymns that praise the Sarasvati River include RV 6.61; 7.96 and 10.17.
Rigveda 7.95.2. and other verses (e.g. 8.21.18) also tell that the Sarasvati poured "milk and ghee." Rivers are often likened to cows in the Rigveda, for example in 3.33.1 cd,
Some Rigvedic verses (6.61.2-13) indicate that the Sarasvati river originated in the hills or mountains (giri), where she "burst with her strong waves the ridges of the hills (giri)". It is a matter of interpretation whether this refers not merely to the Himalayan foothills like the present-day Sarasvati (Sarsuti) river. The Sarasvati is described as a river swollen (pinvamānā) by the rivers (sindhubhih) (RV 6.52.6).
In RV 8.21.18ab mentions a number of petty kings dwelling along the course of Sarasvati,
Another reference to the Sarasvati is in the geographical enumeration of the rivers in the late Rigvedic Nadistuti sukta (10.75.5, this verse enumerates all important rivers from the Ganges in the east up to the Indus in the west in a strict geographical order), as "Ganga, Yamuna, Sarasvati, Shutudri", the Sarasvati is placed between the Yamuna and the Sutlej, consistent with the Ghaggar identification. It is clear, therefore, that even if she has unmistakably lost much of her former prominence, Sarasvati remains characterized as a river goddess throughout the Rigveda.
In RV 3.23.4, the Sarasvati River is mentioned together with the Drsadvati River and the Apaya (Āpayā) River.
In some hymns, the Indus river seems to be more important than the Sarasavati, especially in the Nadistuti sukta. In RV 8.26.18, the white flowing Sindhu 'with golden wheels' is the most conveying or attractive of the rivers.
The name Sarasvati already in the Rigveda does not always relate to a river and its personification exclusively; and in some hymns, the goddess Saraswati (the Hindu goddess of knowledge) is becoming abstracted from the river.
In the 1 and 10 of the Rigveda, the Sarasvati is mentioned in 13 hymns (1.3, 13, 89, 164; 10.17, 30, 64, 65, 66, 75, 110, 131, 141). Only two of these references are unambiguously to the river, 10.64.9 calling for the aid of three "great rivers", Sindhu, Sarasvati and Sarayu, and the geographical Nadistuti list (10.75.5) discussed above. The others invoke Sarasvati as a goddess without direct connection to a specific river. In 10.30.12, her origin as a river goddess may cause the rishi invokes her as protective deity as he composes a hymn to the celestial waters. Similarly, in 10.135.5, as Indra drinks Soma he is described as refreshed by Sarasvati. The invocations in 10.17 address Sarasvati as a goddess of the forefathers as well as of the present generation. In 1.13, 1.89, 10.85, 10.66 and 10.141, she is listed with other gods and goddesses, not with rivers. In 10.65, she is invoked together with "holy thoughts" (dhī) and "munificence" (puraṃdhi), consistent with her role as the goddess of both knowledge and fertility.
In post-Rigvedic literature, Vinasana (the place of disappearance of the Sarasvati), is mentioned. Plaksa Prasravana denotes the place where the Sarasvati appears. In the Rigveda Sutras, Plaksa Prasravana refers to the source of the Sarasvati.
The Vajasaneyi-Samhita of the Yajurveda 34.11 says: "Five rivers, with their flow, go to the Sarasvati. The Sarasvati however became a fivefold stream in the land." The medieval commentator Uvata wrote that the five tributaries of the Sarasvati were the Punjab rivers Drishadvati, Satudri (Sutlej), Chandrabhaga (Chenab), Vipasa (Beas) and the Iravati (Ravi). According to V. S. Wakankar and Parchure, "the five mouths can be identified at Jaisalmer/Badmer. It is significant to note that dried-up remnants of the following five rivers are presently observable near the holy place called Panchabhadra..." 
The Atharva Veda (6.30.1) and some later texts (Taittiriya Brahmana 22.214.171.124, Sutras) say that farming of barley (yava) 'combined with honey' was practiced on the banks of the Sarasvati River.
The first reference to a drying up of the Sarasvati is from the Brahmanas, texts that are composed in Vedic Sanskrit, but dating to a later date than the Veda Samhitas. The Jaiminiya Brahmana (2.297) speaks of the 'diving under (upamajjana) of the Sarasvati', and the Tandya Brahmana calls this the 'disappearance' (vinasana). The same text (25.10.11-16) records that the Sarasvati is 'so to say meandering' (kubjimati) as it could not sustain heaven which it had propped up. The distance between the Plaksa Prasravana (place of appearance/source of the river) and the Vinasana (place of disappearance of the river) is said to be 44 asvina (between several hundred and 1600 miles) (Tandya Br. 25.10.16; cf. Av. 6.131.3; Pancavimsa Br.). 
Both 19th and 20th century fieldwork (Marc Aurel Stein and recent satellite imagery suggest that the Ghaggar-Hakra river in the undetermined past had the Sutlej and the Yamuna as its tributaries. Geological changes diverted the Sutlej towards the Indus and the Yamuna towards the Ganga, and the formerly great river (the Rann of Kutch is likely the remains of its delta) did not have enough water to reach the sea anymore and dried up in the Thar desert. This change is estimated by geologists to have occurred between 5000 and 3000 BC, that is, before the Mature Harappan period. It is sometimes proposed that the Sarasvati of the early Rigveda corresponds to the Ghaggar-Hakra before these changes took place (the "Old Ghaggar"), and the late Vedic end Epic Sarasvati disappearing in the desert to the Ghaggar-Hakra following the diversion of Sutlej and Yamuna, but the 4th millennium date of the event far predates even high estimates of the age of the Rigveda.
The identification of the Vedic Sarasvati River with the Ghaggar-Hakra River was already accepted by Christian Lassen and Max Müller and Marc Aurel Stein. However, an alternate view has located the early Sarasvati River in Afghanistan. The identity of the dried-up Ghaggar-Hakra with the late Vedic and post-Vedic Sarasvati is widely accepted. The identification of the early Rigvedic Sarasvati with the Old Ghaggar is another matter, and the subject of dispute. Kochhar (1999) lists a number of reasons conflicting with the identification:
Suggestions for the identity of the early Rigvedic Sarasvati River include the Helmand River in Afghanistan, separated from the watershed of the Indus by the Sanglakh Range. The Helmand historically besides Avestan Haetumant bore the name Haraxvaiti, which is the Avestan form corresponding to Sanskrit Sarasvati. The Old Persian form is Hara[h]uvati, in Achaemenid times the name of the Arghandab River, the chief tributary of the Helmand. This name was in turn Hellenized to Arachosia. The 1st century CE geographer Isidore of Charax referred to Arachosia, the land where the Arghandab (Sarasvati) and Helmand (Setumanta) flow, as White India.
The Avesta extols the Helmand in similar terms to those used in the Rigveda with respect to the Sarasvati: "the bountiful, glorious Haetumant swelling its white waves rolling down its copious flood" (Yasht 10.67). Kocchar (1999) argues that the Helmand is identical to the early Rigvedic Sarasvati of suktas 2.41, 7.36 etc., and that the Nadistuti sukta (10.75) was composed centuries later, after an eastward migration of the bearers of the Rigvedic culture to the western Gangetic plain some 600 km to the east. The Sarasvati by this time had become a mythical "disappeared" river, and the name was transferred to the Ghaggar which disappeared in the desert, which under the influence of the early hymns was made into an invisible river joining the Ganga and Yamuna.
The possibility of an inverse transfer of the name from India to Iran is proposed by several scholars, who argue that "it would be just as plausible to assume that Sarasvati was a Sanskrit term indigenous to India and was later imported by the speakers of Avestan into Iran."  A transfer of the name from India to Iran, would have taken place in pre-Proto-Iranian times, since the initial *s was regularly changed to h- in proto-Iranian. 
Criticism of the Helmand identification with early Rig Vedic Sarasvati typically points out that the Helmand flows into a swamp in the Iranian plateau (the extended wetland and lake system of Hamun-i-Helmand), which allegedly does not match the Rigvedic description of samudra meaning ocean.
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Thousand of years before, Sarasvati River went below the ground. Now, there is no such river.