Elliptic curves were first introduced to cryptography in the 80’s, and since then they have become the standard for public-key cryptography in research and industry. Current applications range from efficient instantiation of the Diffie-Hellman key exchange to the zero-knowledge proofs underpinning blockchains. This talk covers the basics of Elliptic Curve Cryptography from a practical point of view, highlighting the algorithmic progress throughout the years. Special focus will be given to engineering considerations (efficiency and implementation security) and the many contributions by Latin American cryptographers. We finish by summarizing some applied research problems that hopefully illustrate just how wonderful the topic is.