In the globalization era, cryptography becomes more popular and powerful; in fact it is very important in many areas (i.e. mathematics, computer science, networks, etc). This thesis provides an overview and comparison between the RSA cryptosystem and elliptic curve cryptography, which both focus on sending and receiving messages. The basic theories of the RSA cryptosystem and elliptic curve cryptography are explored. The RSA cryptosystem and elliptic curve cryptography theories are quite similar but elliptic curve cryptography is more complicated. The idea of the RSA cryptosystem is based on three popular theorems which are Euler's Theorem, Fermat's Little Theorem and the Chinese Remainder Theorem. This discussion shows that the reliability and strong security of the RSA cryptosystem depends on the degree of dif- ficulty of integer factorization. Therefore, methods for integer factorization are discussed. In addition I show how the security of elliptic curve cryptography depends on the apparent difficulty of solving the elliptic curve discrete logarithm problem (ECDLP).