The emergence of high-mobility, colloidal semiconductor quantum dot (QD) solids has triggered fundamental studies that map the evolution from carrier hopping through localized quantum-confined states to band-like charge transport in delocalized and hybridized states of strongly coupled QD solids, in analogy with the construction of solids from atoms. Increased coupling in QD solids has led to record-breaking performance in QD devices, such as electronic transistors and circuitry, optoelectronic light-emitting diodes, photovoltaic devices and photodetectors, and thermoelectric devices. Here, we review the advances in synthesis, assembly, ligand treatments and doping that have enabled high-mobility QD solids, as well as the experiments and theory that depict band-like transport in the QD solid state. We also present recent QD devices and discuss future prospects for QD materials and device design.