We present a unified approach to goodness-of-fit testing in R d and on lower-dimensional manifolds embedded in R d based on sums of powers of weighted volumes of k th nearest neighbor spheres. We prove asymptotic normality of a class of test statistics under the null hypothesis and under fixed alternatives. Under such alternatives, scaled versions of the test statistics converge to the α-entropy between probability distributions. A simulation study shows that the procedures are serious competitors to established goodness-of-fit tests. The tests are applied to two data sets of gamma-ray bursts in astronomy.