This thesis studies homomorphic encryption which allows specific types of computations to be carried out on cyphertexts and generate an encrypted result which, when decrypted, matches the result of operations performed on the plaintexts. It may be divided into three fundamental parts. The first part includes the first chapters where we have defined homomorphism and homomorphic encryption. In the second part we describe partially homomorphic cryptosystems which allows only one operation is possible (usually addition or multiplication in the plaintext space) to be able to manipulate the plaintext by using only the cyphertext. There is a lot of partially homomorphic cryptosystems, but in this thesis processed are RSA, ElGamal and Paillier’s cryptosystems. Every encryption scheme consists of the same components: the key generation, the encryption algorithm, the decryption algorithm, homomorphic property and security. At the end, each cryptosystem is confirmed with example which illustrate their execution. The third part of the thesis refers to the fully homomorphic encryption which allows one to evaluate both addition and multiplication of plaintext, while remaining encrypted. First we described fully homomorphic encryption in general and introduce some of its properties. Then we gave a historical overview. At the end we showed connection between simple somewhat encryption scheme and fully homomorphic encryption scheme with all the advantages and disadvantages.