During the talk, I will present analytical and numerical results for the (possibly nonlinear) coupled equations of poroelasticity describing the fluid flow in a deformable porous medium. We will focus on novel schemes based on a Hybrid High-Order discretization of the mechanics and a Symmetric Weighted Interior Penalty discontinuous Galerkin discretization of the flow. The method has several assets, including, in particular, the validity in two and three space dimensions, inf-sup stability, and the support of general polyhedral meshes, nonmatching interfaces, and arbitrary space approximation order. Our analysis delivers stability and error estimates that hold also when the constrained specific storage coefficient vanishes, and shows that the constants have only a mild dependence on the heterogeneity of the permeability coefficient. The performance of the method is extensively investigated on a complete panel of model problems using stress-strain laws corresponding to real materials. In the last part of the talk, we will consider the numerical solution of the poroelasticity problem with random physical coefficients in the context of uncertainty quantification. The uncertainty is modelled with a finite set of parameters with prescribed probability distribution. The approximation of the stochastic partial differential system is realized by non-intrusive techniques based on polynomial chaos decompositions. We will conclude by performing a sensitivity analysis to asses the propagation of the input uncertainty on the solutions considering application-oriented test cases.