Neurons often respond to diverse combinations of task-relevant variables. This form of mixed selectivity plays an important computational role which is related to the dimensionality of the neural representations: high-dimensional representations with mixed selectivity allow a simple linear readout to generate a huge number of different potential responses. In contrast, neural representations based on highly specialized neurons are low dimensional and they preclude a linear readout from generating several responses that depend on multiple task-relevant variables. Here we review the conceptual and theoretical framework that explains the importance of mixed selectivity and the experimental evidence that recorded neural representations are high-dimensional. We end by discussing the implications for the design of future experiments.