A general greedy approach to construct coverings of compact met-ric spaces by metric balls is given and analyzed. The analysis is a continuousversion of Chv´atal’s analysis of the greedy algorithm for the weighted set coverproblem. The approach is demonstrated in an exemplary manner to constructefficient coverings of the n-dimensional sphere and n-dimensional Euclideanspace to give short and transparent proofs of several best known bounds ob-tained from deterministic constructions in the literature on sphere coverings