We develop methods to incorporate paleogeographical constraints into numerical models of mantle convection. Through the solution of the convection equations, the models honor geophysical and geological data near the surface while predicting mantle flow and structure at depth and associated surface deformation. The methods consist of four constraints determined a priori from a plate history model: (1) plate velocities, (2) thermal structure of the lithosphere, (3) thermal structure of slabs in the upper mantle, and (4) velocity of slabs in the upper mantle. These constraints are implemented as temporally- and spatially-dependent conditions that are blended with the solution of the convection equations at each time step. We construct Earth-like regional models with oceanic and continental lithosphere, trench migration, oblique subduction, and asymmetric subduction to test the robustness of the methods by computing the temperature, velocity, and buoyancy flux of the lithosphere and slab. Full sphere convection models demonstrate how the methods can determine the flow associated with specific tectonic environments (e.g., back-arc basins, intraoceanic subduction zones) to address geological questions and compare with independent data, both at present-day and in the geological past (e.g., seismology, residual topography, stratigraphy). Using global models with paleogeographical constraints we demonstrate (1) subduction initiation at the Izu-Bonin-Mariana convergent margin and flat slab subduction beneath North America, (2) enhanced correlation of model slabs and fast anomalies in seismic tomography beneath North and South America, and (3) comparable amplitude of dynamic and residual topography in addition to improved spatial correlation of dynamic and residual topography lows.