The goal of this project is to get familiar with the Learning With Errors (LWE) cryptosystem. LWE is the basis for many of the most prominent post-quantum encryption algorithms. Conceptually, the logic behind the secrecy of LWE is based on representing a secret as a set of linear equations containing noise. This problem can also be viewed as a lattice problem (namely shortest vector problem (SVP)), which is known to be NP-hard and there is no known polynomial time quantum algorithm either. In the report, I will introduce the basic concepts and math behind LWE, and describe a simple public key system utilizing the LWE problem.