In this talk we review some recent developments on space-time methods based on well-posed formulations of time-dependent PDEs. While variational formulations of elliptic PDEs often use the same solution and test space, this is different for time-dependent PDEs such as the heat equation and the wave equation. For the latter, the correct functional analytic setting is not clear in the literature. We consider the first-order system of the acoustic wave equation and prove that the operator corresponding to this system is an isomorphism from an appropriately defined graph space to the space of square-integrable functions. This novel well-posedness result paves the way to defining finite element methods of least-squares type on the space-time domain. The analysis uses stability results of weak and ultra-weak formulations of the second-order wave equation. Some numerical examples and discussion on future research conclude this talk. This is a joint work with Roberto Gonz?lez and Michael Karkulik from Universidad T?cnica Federico Santa Mar?a, Valpara?so, Chile and Gregor Gantner from University of Bonn, Germany.