Colloidal quantum dots (CQDs) are solution-processed semiconductors of interest in low-cost photovoltaics. Tuning of the bandgap of CQD films via the quantum size effect enables customization of solar cells' absorption profile to match the sun's broad visible-and infrared-containing spectrum reaching the earth. Here we review recent progress in the realization of low-cost, efficient solar cells based on CQDs. We focus in particular on CQD materials and approaches that provide both infrared and visible-wavelength solar power conversion CQD photovoltaics now exceed 5% solar power conversion efficiency, achieved by the introduction of a new architecture, the depleted-heterojunction CQD solar cell, that jointly maximizes current, voltage, and fill factor. CQD solar cells have also seen major progress in materials processing for stability, recently achieving extended operating lifetimes in an air ambient. We summarize progress both in device operation and also in gaining new insights into materials properties and processing - including new electrical contact materials and deposition techniques, as well as CQD synthesis, surface treatments, film-forming technologies - that underpin these rapid advances.