I will present some old and new results about the L^p functional calculus for sub-Laplacians L. It has been known for a long time that, under fairly general assumptions on the
sub-Laplacian and the underlying sub-Riemannian structure, an operator of the form F(L) is bounded on L^p (1<p<infinity) whenever the spectral multiplier F satisfies a
scale-invariant smoothness condition of sufficiently large order. The problem of determining the minimal smoothness assumptions, however, remains widely open and is intimately
connected with that of proving sharp estimates for the sub-Riemannian wave equation.