Computational Linguistics is an interdisciplinary field of computer science and linguistics that focuses on designing computational models and algorithms for processing, analyzing, and generating human language. Over recent years, this field has made substantial progress. While its primary emphasis tends to center around widely spoken languages, there is equal importance in investigating languages that are not commonly spoken but have contributed immensely to the literature, culture, and philosophy of the society. Thus, this survey article comprehensively delves into the exploration of computational tasks undertaken for Sanskrit, an ancient language of the Indian sub-continent steeped in a wealth of literary heritage. The purpose of this study is to provide an overview of the progress made thus far in the computational analysis of Sanskrit, while also reviewing the current digital infrastructure that supports these efforts. Additionally, our study also identifies potential avenues for future research, serving as a reference for anyone interested in advancing their exploration in this field.