Abstract. 
A Lyapunov function is a a function which is non-increasing along orbits
of a dynamical systems. Continuous Lyapunov functions for flows have been
constructed by Conley. Smooth Lyapunov functions can be constructed by
Conley's method for homeomorphisms. However there are examples of
continuous flows which do not admit a smooth non-constant Lyapunov
function. We will explain this phenomenon. We will also give results on
the possibility of approximating a Lyapunov function by a smooth Lyapunov
function.