The author studies the approximation behaviour of the spherical partial sums of double Fourier series of functions in the two-dimensional Waterman classes [see \textit{D. Waterman}, Stud. Math. 44, 107-117 (1972; Zbl 0207.06901)] for the one-dimensional case. He also considers the problem of convergence of the spherical partial sums of double Fourier series of the characteristic functions of convex sets on the two-dimensional torus. The results obtained are too lengthy to be reproduced here. At the end, the author presents an example which shows that analogous results are no longer true in the three-dimensional case.