I will start with a self-contained introduction to the homogenization of inviscid (first-order) and viscous (second-order) Hamilton-Jacobi (HJ) equations in stationary ergodic media in any dimension. After a brief account of the now-classical works that are concerned with periodic media or convex Hamiltonians, I will return to the general setting and outline the results obtained in the last decade that: (i) established homogenization for inviscid HJ equations in one dimension; and (ii) provided counterexamples to homogenization in the inviscid and viscous cases in dimensions two and higher. Finally, I will present my recent joint work with E. Kosygina in which we prove homogenization for viscous HJ equations in one dimension, and also describe how the solution of this problem qualitatively differs from that of its inviscid counterpart.