I will start the talk by recalling the notion of gradient flow in its easiest occurrence: the evolution equation x'(t)=-grad F(x(t)) in the Euclidean space. In particular, the focus will be on the implicit Euler scheme as a sequence of iterated minimization problems. I will then move to a more involved setting, where the point x is replaced by a probability density ρ evolving in the space of probabilities endowed with the so-called Wasserstein distance, induced by optimal transport. For suitable choices of the functional F one can recover linear diffusion PDEs (heat and Fokker-Planck equations) as well as non-linear ones (porous medium, fast diffusion, models for crowd motion). The iterated minimization scheme is called in this case JKO scheme (from Jordan-Kinderlehrer-Otto). After explaining why this scheme heuristically provide the desired equation at the limit, I will show how its optimality conditions can be exploited to prove estimates on its solutions, in particular BV, Sobolev and Lipschitz bounds. Lipschitz estimates can also be interpreted as bounds on the maximal displacement of each particle in the optimal transport map, and have a numerical interest, which I will discuss in two examples, where a potential drift is coupled either with linear diffusion or with a pressure effect due to density constrained in crowd motion.