Finding the sub-vector with the largest sum in a sequence of n numbers is known as the maximum sum problem. Finding the k sub-vectors with the largest sums is a natural extension of this, and is known as the k maximal sums problem. In this paper we design an optimal O(n + k) time algorithm for the k maximal sums problem. We use this algorithm to obtain algorithms solving the two-dimensional k maximal sums problem in O(m ²·n + k) time, where the input is an m ×n matrix with m ≤ n. We generalize this algorithm to solve the d-dimensional problem in O(n ^2d − 1 + k) time. The space usage of all the algorithms can be reduced to O(n ^d − 1 + k). This leads to the first algorithm for the k maximal sums problem in one dimension using O(n + k) time and O(k) space.