Solution processed semiconductors are in demand for present and next-generation optoelectronic technologies ranging from displays to quantum light sources because of their scalability and ease of integration into devices with diverse form factors. One of the central requirements of semiconductors used in all these applications is a narrow photoluminescence (PL) linewidth. Narrow emission linewidths are needed to ensure both color and single-photon purity, raising the question of what design rules are needed to obtain narrow emission from semiconductors made in solution. In this review we first examine the requirements for colloidal emitters for a variety of applications including light-emitting diodes, photodetectors, lasers, and quantum information science. Next, we will delve into the sources of spectral broadening, including homogeneous broadening from dynamical broadening mechanisms in single-particle spectra, heterogeneous broadening from static structural differences in ensemble spectra, and spectral diffusion. Then, we compare the current state of the art in terms of emission linewidth for a variety of colloidal materials including II-VI quantum dots (QDs) and nanoplatelets, III-V QDs, alloyed QDs, metal-halide perovskites including nanocrystals and 2D structures, doped nanocrystals, and, finally, as a point of comparison, organic molecules. We end with some conclusions and connections, with an outline of promising paths forward.