Cosmological inflation is one of the leading paradigms for explaining the physical conditions that prevailed in the early Universe. It consists in a phase of very high energy accelerated expansion that solves the hot big bang model problems. When combined with quantum mechanics, it also provides a causal mechanism for generating cosmological fluctuations on large scales.
In this talk I will explain how future CMB missions will allow one to extract a few favoured models among hundreds of currently proposed scenarios. I will insist on the role played by reheating, which connects inflation to the subsequent radiation era and during which ordinary matter as we see it today is produced; and show how this epoch will also be constrained. I will finally discuss some extensions to the minimal single-field setups, where extra scalar fields are added and play a role both during inflation and reheating.