Metal halide perovskites are not only excellent materials for photovoltaic and light-emitting applications — they are also an intriguing riddle for researchers trying to understand the relationship between structure and optoelectronic properties in these semiconductors. A very broad range of experimental techniques, as well as computer simulations, have been used to extract key charge transport parameters, such as diffusion lengths, lifetimes, recombination rates and effective masses of the charge carriers, sometimes leading to behaviour unexpected for solution-made, inherently soft materials. We asked four teams of researchers to share their views on the current understanding of the photophysics of perovskites. They discuss some of the properties and effects — such as structural fluctuations, lattice softness, ferroelectric-like large polarons, Rashba effects and defect photochemistry — that may make these semiconductors different from other materials used in optoelectronics. Moreover, they suggest further investigations to clarify the relative impact of these properties on the performance of solar cells, and propose ways to exploit some of these effects in other devices based on metal halide perovskites.