We discuss the relation between kinetic formulation, the notion of nonlinearity and the regularizing effects of first- and second-order equations. In particular, we highlight the regularizing effect of nonlinear entropy solution operators of quasilinear conservation laws and compare it to recent 2D results based on compensated compactness arguments. We also discuss new regularity estimates for second-order equations with possibly non-isotropic degenerate elliptic part.