Several interesting asymptotic properties of Hamilton-Jacobi equations are based on the so-called critical value of the Hamiltonian H(x,p) and on the associated critical stationary H-J equation. In particular, the long time behaviour of evolutive H-J equations is described in terms of the critical value and a critical solution, and so is the homogenisation of H-J equations with highly oscillating ingredients. The theory was  pioneered by Lions, Papanicolaou and Varadhan and by A. Fathi, and it has applications to ergodic control and to dynamical systems, the so-called weak KAM theory. Most of the known result assume the coercivity of the Hamiltonian in the moment variables p, and interpret the critical equation in the sense of continuous viscosity solutions. After reviewing some classical results, I will present some recents improvements holding for non-coercive Hamiltonians arising from the optimal control of affine systems, possibly with an uncontrolled drift term. Different from the previous theory, I use in a crucial way viscosity solutions that can be discontinuous.