How do proteins evolve? How do changes in sequence mediate changes in protein structure, and in turn in function? This question has multiple angles, ranging from biochemistry and biophysics to evolutionary biology. This review provides a brief integrated view of some key mechanistic aspects of protein evolution. First, we explain how protein evolution is primarily driven by randomly acquired genetic mutations and selection for function, and how these mutations can even give rise to completely new folds. Then, we also comment on how phenotypic protein variability, including promiscuity, transcriptional and translational errors, may also accelerate this process, possibly via "plasticity-first" mechanisms. Finally, we highlight open questions in the field of protein evolution, with respect to the emergence of more sophisticated protein systems such as protein complexes, pathways, and the emergence of pre-LUCA enzymes.