Given two sets A and B of multidimensional objects, the all-nearest-neighbors (ANN) query retrieves for each object in A its nearest neighbor in B. Although this operation is common in several applications, it has not received much attention in the database literature. In this paper we study alternative methods for processing ANN queries depending on whether A and B are indexed. Our algorithms are evaluated through extensive experimentation using synthetic and real datasets. The performance studies show that they are an order of magnitude faster than a previous approach based on closest-pairs query processing.