Upconversion of sunlight with energy below the band gap of a solar cell is a promising technique for enhancing the cell efficiency, simply by utilizing a larger part of the solar spectrum. The present topical review addresses this concept and discusses the material properties needed for an efficient upconversion process with focus on both silicon and organic solar cells. To design efficient upconverters, insight into topics such as quantum-optics, nano-optics, numerical modeling, optimization, material fabrication, and material characterization is paramount, and the necessary concepts are introduced throughout the review. Upconversion modeling is done using rate equations, while optical modeling is done by solving Maxwell's equations using the finite element method. Topology optimization is introduced and used to generate geometries of gold nanoparticles capable of greatly enhancing the upconversion yield. Fabrication and experimental characterization methods are discussed. Some recent results are presented and finally the possibility of designing upconverting materials capable of increasing the short-circuit current in a solar cell is discussed.