The three-dimensional, parabolized form of the coupled equations of motion and energy are solved to study the development of mixed-convection heat transfer in the entrance region of horizontal square ducts. The specific problem considered here is that of axially uniform heat flux and peripherally uniform temperature with parabolic inlet profile and uniform inlet temperature in a square cross-section. Previous studies have been confined mostly to two-dimensional flows in the fully developed region. Recent two-dimensional calculations have indicated the existence of multiple steady-state solutions in the fully developed region with two- and four-cell flow structure. Present three-dimensional calculations, carried over the whole entrance length and the full cross-section, indicate the evolutionary path that lead to such two-dimensional flows. Furthermore they shed new light on the nature of flows in regions where all known two-dimensional solutions become unstable in some manner. For low Grashof numbers and Pr = 0.73 the secondary velocities develop into an axially invariant state with two counter-rotating vortices. For Grashof numbers above 2.2 x 105 the inlet profiles evolve into a state with a four-cell secondary flow structure. As in a related problem of flow in a curved channel (Winters 1987; Bara et al. 1992), the two-dimensional, four-cell solutions are found to be unstable to asymmetric perturbations. Such perturbations trigger a new, streamwise-periodic mode which is sustained over long lengths in the flow direction. For Grashof numbers above 5.6 x 105 axially invariant two-cell solutions reappear. Some of the three-dimensional solutions corresponding to the streamwise periodic mode lack the reflective symmetry about the vertical centreline and hence these flows occur with multiplicity of two.