Nonclassical symmetry methods are used to study the nonlinear diffusion equation with a nonlinear source. In particular, exponential and power law diffusivities are examined and we obtain mathematical forms of the source term which permit nonclassical symmetry reductions. In addition to the known source terms obtained by classical symmetry methods, we find new source terms which admit symmetry reductions. We also deduce a class of nonclassical symmetries which are valid for arbitrary diffusivity and deduce corresponding new solution types. This is an important result since previously only traveling wave solutions were known to exist for arbitrary diffusivity. A number of examples are considered and new exact solutions are constructed.