We will explore recent advances concerning nonlinear diffusion processes in the sense of McKean-Vlasov, and their connections to partial differential equations (PDEs) defined on the Wasserstein space, that is, the space of probability measures with finite second order moment. We will discuss recent results on the well-posedness - both in the weak and strong sense - of McKean-Vlasov stochastic differential equations (SDEs) driven by Brownian motion and/or jump processes. These results extend beyond the classical Cauchy-Lipschitz framework. In the Brownian setting, we will describe the regularization effect of the noise, notably the existence and smoothness of the transition density - particularly in the measure argument - under uniform ellipticity assumptions. These smoothing effects are crucial for establishing the existence and uniqueness of solutions to the Kolmogorov-type PDEs posed on the Wasserstein space, even in the presence of irregular terminal conditions and source terms. Such infinite-dimensional PDEs play a central role in deriving quantitative propagation of chaos estimates for mean-field approximations via interacting particle systems. Finally, we will discuss the numerical approximation of these equations using the Euler-Maruyama time discretization scheme.