In this course we will present the main concepts and results of the so-called theory of viscosity solutions which applies to fully nonlinear 1st and 2nd order PDEs. For such equations, solutions generally are nonsmooth and standard approaches do not apply: weak, strong, measure-valued and distributional solutions either do not exist or can not even be defined. The name of this theory is after the "vanishing viscosity method" where it first originated, but now constitutes an independent theory of "weak" solutions which applies to nondivergence form PDEs. Interesting PDEs arise for example in Geometry and Geometric Evolution (Monge-Ampere PDEs, Motion by Mean Curvature, Optimal Lipschitz Extensions), Optimal Control and Game Theory (Hamilton-Jacobi-Bellman PDEs, Isaacs PDEs) and Calculus of Variations in Lp and L∞ (Euler-Lagrange PDEs, $p$-Laplacian, Aronsson PDEs, ∞-Laplacian). We will focus in particular on applications in Calculus of Variations in L∞. No previous knowledge will be assumed for the audience and basic graduate-level mathematical maturity will suffice for the main core of this course.