The lectures will start by reviewing closed topological theories, before moving to more recent work in the open/closed theory on spaces with boundary. From the work of Witten, the topology of the so-called Deligne-Mumford moduli spaces of Riemann surfaces with nodes plays a fundamental role in 2-dimensional topological gravity (known to mathematicians as Gromov-Witten theory). For example, by the work of Kontsevich and Manin, it is seen to underly the Witten-Dijkgraaf-Verlinde-Verlinde equation, and hence is intimately related to the theory of Frobenius manifolds and of integrable systems. Most work on these moduli spaces has been focussed on the case of closed topological field theory. In these lectures, I will explore the moduli spaces, analogous to Deligne-Mumford moduli spaces, which play the corresponding role in the open theory. In this case, the world sheet (Riemann surface) has a boundary, and as a result, the moduli spaces are no longer complex orbifolds, but rather real orbifolds with corners. These moduli spaces may be viewed as an explanation of the way that algebraic structures, such as A-infinity categories, cyclic homology, and the Cardy condition, enter topological field theory in two dimensions. This theory should also have applications to understanding the foundations of string theory.