The basic reproduction number R0 is an important concept in population biology. As a threshold quantity for population dynamics, it is unquestionably one of the most valuable mathematical ideas brought to theoretical ecology and epidemiology. In this talk, I will report our recent research on the linear stability and R0 for autonomous and cooperative functional differential equations (FDEs). In particular, we will give a general formula of R0 for autonomous and compartmental FDE models, where both the production (infection) and internal transition have time delays. We also extend this theory to abstract autonomous FDEs on an ordered Banach space so that it can be applied to time-delayed reaction-diffusion systems. As an illustrative example, we establish the threshold dynamics for a time-delayed population model of black-legged ticks in terms of R0.