Representational models explain how activity patterns in populations of neurons (or, more generally, in multivariate brain activity measurements) relate to sensory stimuli, motor actions, or cognitive processes. In an experimental context, representational models can be defined as probabilistic hypotheses about what activation profiles across experimental conditions are likely to be observed. We describe three methods to test such models - encoding approaches, pattern component modeling (PCM), and representational similarity analysis (RSA). We show that these methods are closely related in that they evaluate the statistical second moment of the activity profile distribution. Using simulated data from three different fMRI experiments, we compare the power of the approaches to adjudicate between competing representational models. PCM implements a likelihood-ratio test and therefore constitutes the most powerful test if its assumptions hold. However, the other two approaches - when conducted appropriately - can perform similarly. In encoding approaches, the linear model needs to be appropriately regularized, which imposes a prior on the activity profiles. Without such a prior, encoding approaches do not test well-defined representational models. In RSA, the unequal variances and dependencies of the distance measures need to be taken into account to enable near-optimal inference. The three techniques render different aspects of the information explicit (e.g. single response tuning in encoding approaches and population representational dissimilarity in RSA) and have specific advantages in terms of computational demands, ease of use, and extensibility. We argue that they constitute complementary parts of the same computational toolkit aimed at understanding neural representations on the basis of multivariate brain-activity data.