\indent In the last 25 years, Elliptic Curve Cryptography has become a mainstream primitive for cryptographic protocols and applications. This talk will give a survey of elliptic curve cryptography and its applications, including applications of pairing-based cryptography which are built with elliptic curves. One of the information-theoretic applications I will cover is a solution to prevent pollution attacks in content distribution networks such as BitTorrent which use network coding to achieve optimal throughput. Another application I will cover is to privacy of electronic medical records and billings systems.