The concept of Fully Homomorphic Encryption (FHE) was believed to be the ultimate goal of cryptography until 2009 when Gentry presented the first FHE scheme. FHE allows multiple operations on plaintext without the need for decryption, hence enable arbitrary complex computation on encrypted data. It addresses privacy concerns in different fields, such as machine learning, cloud computing and fog computing. In order to achieve this goal both addition and multiplication must be executed homomorphically since any Boolean (arithmetic) circuit can be represented by XOR (addition) and AND (multiplication). This thesis work includes the explanation of FHE from theoretical definitions and security properties to the possible use cases in which it can be applied.