  The theory of traveling waves and spreading speeds is developed for
time-space periodic monotone semiflows with monostable structure. By using
traveling waves of the associated Poincar\'e maps in a strong sense, we
establish the existence of time-space periodic traveling waves and spreading
speeds. We then apply these abstract results to a two species competition
reaction-advection-diffusion model. It turns out that the minimal wave speed
exists and coincides with the single spreading speed for such a system no
matter whether the spreading speed is linearly determinate. We also obtain a
set of sufficient conditions for the spreading speed to be linearly
determinate.
