Propagation has been modelled by reaction-diffusion equations since the pioneering works of Fisher and Kolmogorov-Peterovski-Piskunov (KPP). Much new developments have been achieved in the past a few decades on the modelling of propagation, with traveling wave and related solutions playing a central role. In this talk, I will report some recent results obtained with several collaborators on some reaction-diffusion models with free boundary and "nonlocal diffusion", which include the Fisher-KPP equation (with free boundary) and two epidemic models. A key feature of these problems is that the propagation may or may not be determined by traveling wave solutions. There is a threshold condition on the kernel functions which determines whether the propagation has a finite speed or infinite speed (known as accelerated spreading). For some typical kernel functions, we obtain sharp estimates of the spreading speed (whether finite or infinite).