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The Sanskrit grammatical tradition of vyākaraṇa is one of the six Vedanga disciplines. It has its roots in late Vedic India, and includes the famous work, Aṣṭādhyāyī, of Pāṇini (ca. 5th century BC).
The impetus for linguistic analysis and grammar in India originates in the need to be able to obtain a strict interpretation for the Vedic texts. The work of the very early Indian grammarians have been lost; for example, the work of Sakatayana (ca. 8th c. BC) is known only from cryptic references by Yaska (ca. 7th c. BC) and Panini. One of the views of Sakatayana that was to prove controversial in coming centuries was that most nouns are etymologically derivable from verbs.
In his monumental work on etymology, Nirukta, Yaska supported this claim based on the large number of nouns that were derived from verbs through a derivation process that became known as krit-pratyaya; this relates to the nature of the root morphemes.
Yaska also provided the seeds for another debate, whether textual meaning inheres in the word (Yaska's view) or in the sentence (see Panini, and later grammarians such as Prabhakara or Bhartrihari). This debate continued into the 14th and 15th c. AD, and is relevant even today perhaps, with the debate on the Dynamic Turn in Semantics, which says that meaning in language is dynamically created and it may not be possible to compose the meaning from those of the words.

Medieval Accounts

Modern Sanskrit grammarians

Jean Francois Pons
Henry Thomas Colebrooke
August Wilhelm von Schlegel
Wilhelm von Humboldt
Dimitrios Galanos 19th century

Bernhard Geiger
Leonard Bloomfield
Paul Thieme
Louis Renou
Herman Buiskool
Bimal Krishna Matilal
Johannes Bronkhorst
George Cardona
Madhav Deshpande
SD Joshi
Paul Kiparsky
Frits Staal
Michael Witzel
Kshetresa Chandra Chattopadhyaya
Vagish Shastri
Sri Sribhuti Krishna Goswami
Sri Pundrik Goswami