Due to the efforts of Fermat, Euler, Legendre and Gauss, it is known what natural numbers can be written as the sum of two squares or three squares. Lagrange's four-square theorem proved in 1770 states that each natural number can be expressed as the sum of four squares.

In the talk we will first review classical results on sums of two or three or four squares, and then turn to the speaker's recent discoveries which refine Lagrange's four-square theorem or the Gauss-Legendre theorem on sums of three squares. We will mainly introduce our results refining Lagrange's four-square theorem as well as some recent conjectures of the speaker one of which states that any integer greater than one can be written as the sum of two squares, a power of 3 and a power of 5.