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 which use network coding to achieve optimal throughput. One solution is based on a pairing-based signature scheme using elliptic curves. This scheme has since been improved by many others, including Boneh et al, and Cheon et al.