### Abstract:

The purpose of this thesis is to investigate the problem of representing aninteger as sum of two, three, and four squares. First the necessary and Sufficient conditions for an integer to be representable as the sum of two and four squares are considered. Then I investigate the problem of the representation of integers as a sum of two and four nonvanishing squares. Next the problem of representing integers as a sum of two and four unequal squares is studied. The uniqueness of representations is also be discussed. Formulas for the total number of representation of an integer as a sum of two and four squares are given. For the sum of three squares problem I characterize the integers that can be represented as a sum of three squares, and only give formulas without proofs for the number of representations of an integer as a sum of three squares, since their proofs are beyond the scope of this thesis.