The Cauchy-Liouville theorem (1844) states that any bounded entire function of a complex variable is necessarily constant. In the realm of PDE's, by a Liouville-type theorem, one usually means a statement asserting the nonexistence of solutions in the whole space (or a suitable unbounded domain). Numerous results of this kind have appeared over the years and many far-reaching applications have arisen, conferring Liouville-type theorems an important role in the theory of PDE's and revealing strong connections with other mathematical areas (calculus of variations, geometry, fluid dynamics, optimal stochastic control). After a brief historical detour (minimal surfaces - Lagrange, Bernstein, de Giorgi, Bombieri,… and regularity theory for linear elliptic systems - Giaquinta, Necas, ...), we will recall the developments of the 1980-2000's on nonlinear elliptic problems, leading to powerful tools for existence and a priori estimates for Dirichlet problems (Gidas, Spruck, Caffarelli, ...), based on the combination of Liouville type theorems and renormalization techniques. In a more recent period, this line of research has also led to much progress in the study of singularities of solutions, both for stationary (elliptic) and evolution PDEs. In particular, in the case of power like nonlinearities, we will recall the equivalence between Liouville type theorems and universal estimates, based on a method of doubling-rescaling (joint work with P. Polacik and P. Quittner, 2007). Then we will present recent developments which show that these renormalization techniques can be applied to nonlinearities without any scale invariance, even asymptotically, with applications to initial and final blowup rates or decay rates in space and/or time. Nota: Questo seminario rientra tra le attività del progetto MUR "Dipartimenti d'eccellenza" MatMod@TOV (2023-27)