There are two schools, or lines, of thought which attempt to unify the apparently divergent laws of dynamics and thermodynamics and to explain the observed time asymmetry of the universe and most of its subsystems in spite of the fact that these systems are driven by time-symmetric evolution equations. They will be called the coarse-graining and the extended dynamics schools (even if these names only partially describe their philosophy). The coarse-graining school obtains time asymmetry via a projection of the state space onto a space of “relevant” states. The corresponding projection of the primitive reversible evolution laws yields effective irreversible evolution laws for the relevant states. Extended dynamics always uses the same primitive reversible evolution laws. But these laws (in adequate extensions of the usual spaces where they are formulated) have a set of solutions S that can be decomposed into two subsets S+ and S- of time-asymmetric solutions. Time asymmetry is established by choosing one of these two sets as the arena in which to formulate the theory. This paper explains in the simplest self-contained and unbiased way the main characteristics of both schools and points out the advantages and disadvantages of each, in such a way as to make explicit the debate between the schools. Some cosmological features of the theory are also considered, mainly the problem of the low-entropy initial state of the universe.