Although the nonlinear diffusion equation has been extensively studied and there exists substantial literature in many diverse areas of science and technology, the number of exact concentration profiles is nevertheless limited. In a recent article in this journal (Hill [1]) a brief review of known exact results is given, as well as an elementary integration procedure which appears to be a general device for obtaining integrals associated with similarity solutions. This paper extends the results given in [1] and for particular power law diffusivities c" (such as m = -1/2, -1, -3/2 and -2) presents a number of new exact solutions obtained by fully integrating the ordinary differential equations derived in [1]. In addition new results are found for a general nonlinear diffusion equation which includes one-dimensional diffusion with an inhomogeneous and nonlinear diffusivity c"x" as well as symmetric nonlinear diffusion in cylinders and spheres. Moreover by a separate and ad-hoc procedure a new solution is obtained of the travelling wave type but with a variable wave speed. Some of the new exact solutions obtained for one-dimensional nonlinear diffusion with power law diffusivities c are illustrated graphically.