Abstract: There is a broad literature on the structure of a random partition of a large integer n, with no arithmetic restriction on the summands. In this work we study random partitions into squares, assuming as in the case of ordinary partitions that the space of all partitions of n into squares is equipped with the uniform distribution. Our primary focus will be to determine the limiting probability distributions for the numbers of summands of various sizes, both "small" and "large", as well as the total number of summands.