Given a set of points in a Euclidean space, and a partitioning of this 'training set' into two or more subsets ('classes'), we consider the problem of identifying a 'reasonable' assignment of another point in the Euclidean space ('query point') to one of these classes. The various classifications proposed in this paper are determined by the distances between the query point and the points in the training set. We report results of extensive computational experiments comparing the new methods with two well-known distance-based classification methods (k-nearest neighbors and Parzen windows) on data sets commonly used in the literature. The results show that the performance of both new and old distance-based methods is on par with and often better than that of the other best classification methods known. Moreover, the new classification procedures proposed in this paper are: (i) easy to implement, (ii) extremely fast, and (iii) very robust (i.e. their performance is insignificantly affected by the choice of parameter values).