Multijunction solar cells have proven to be capable of extremely high efficiencies by combining multiple semiconductor materials with bandgaps tuned to the solar spectrum. Reaching the optimum set of semiconductors often requires combining high-quality materials with different lattice constants into a single device, a challenge particularly suited for metamorphic epitaxy. In this article, we describe different approaches to metamorphic multijunction solar cells, including traditional upright metamorphic, state-of-the-art inverted metamorphic, and forward-looking multijunction designs on silicon. We also describe the underlying materials science of graded buffers that enables metamorphic subcells with low dislocation densities. Following nearly two decades of research, recent efforts have demonstrated high-quality lattice-mismatched multijunction solar cells with very little performance loss related to the mismatch, enabling solar-to-electric conversion efficiencies over 45%.