Colloidal suspensions of monodisperse spherical particles have been extensively studied since one of the main advantages of these systems is their similarity to atomic ones. This property has been used successfully in basic science to understand the equilibrium and non-equilibrium behavior of model colloids and to correlate them with their atomic counterparts. In contrast, suspensions used in technological processes are usually more complex. Nevertheless, for their effective applications, it is crucial to understand their properties, such as the microstructure, dynamics, and flow behavior, as well as the mechanisms underlying their self-organization. The first step towards this knowledge is switching from monodisperse suspensions to moderately complex ones, namely binary mixtures. Therefore, the present review aims to summarize the current knowledge about the phase behavior of binary mixtures of spherical colloids with different inter-particle interactions, such as nearly hard spheres, electrostatic repulsion/attraction, depletion attraction, and attraction due to DNA hybridization. A comparison of experimental work with theoretical predictions is described for binary suspensions studied in three and two dimensions. Several open questions are outlined in the conclusions.