Given a nonlinear wave equation whose nonlinearity contains derivatives of the unknown function: $-\partial_t^2\phi+\sum_{i=1}^ 3 \partial_{x_i}^2\phi=N(\partial\phi)$ in $3+1$ dimensions, what can we say about the global existence of solutions? What if the initial conditions are small and compactly supported? If time permits, I will discuss the case of a wave map.