We consider nonlinear diffusion problems of the form with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For any which is and satisfies , we show that every bounded positive solution converges to a stationary solution as . For monostable, bistable and combustion types of nonlinearities, we obtain a complete description of the long-time dynamical behavior of the problem. Moreover, by introducing a parameter in the initial data, we reveal a threshold value such that spreading () happens when , vanishing () happens when , and at the threshold value , is different for the three different types of nonlinearities. When spreading happens, we make use of “semi-waves” to determine the asymptotic spreading speed of the front.