Three decades ago, dye-sensitized solar cells (DSSCs) emerged as a route for harnessing the sun’s energy and converting it into electricity. Since then, an impressive amount of work has been devoted to improving the global photovoltaic efficiency of DSSCs, trying to optimize all components of the device. Up to now, the best efficiencies have usually been reached with ruthenium(II) photosensitizers, even if in the last few years many classes of organic compounds have shown record efficiencies. However, the future of DSSCs is stringently connected to the research and development of cheaper materials; in particular, the replacement of rare metals with abundant ones is an important topic in view of the long-term sustainability of DSSCs intended to replace the consolidated fossil-based technology. In this context, copper is a valid candidate, being both an alternative to ruthenium in the fabrication of photosensitizers and a material able to replace the common triiodide/iodide redox couple. Thus, recently, some research papers have confirmed the great potential of copper(I) coordination complexes as a cheap and convenient alternative to ruthenium dyes. Similarly, the use of copper compounds as electron transfer mediators for DSSCs can be an excellent way to solve the problems related to the more common I3−/I− redox couple. The goal of this mini-review is to report on the latest research devoted to the use of versatile copper complexes as photosensitizers and electron shuttles in DSSCs. The coverage, from 2022 up to now, illustrates the most recent studies on dye-sensitized solar cells based on copper complexes as molecular materials.