Dispersion problems involve arranging a set of points as far away from each other as possible. They have numerous applications in the location of facilities and in management decision science. We present several algorithms and hardness results for dispersion problems using different natural measures of remoteness, some of which have been studied previously in the literature and others which we introduce; in particular, we give the first algorithm with a non-trivial performance guarantee for the problem of locating a set of points such that the sum of their distances to their nearest neighbor in the set is maximized.