We state the Yamabe conjecture and briefly review the techniques that have been used to study the Yamabe problem on Riemannian manifolds with smooth metrics. We review some of the basic tools from functional analysis and the theory of partial differential equations for studying the problem in the case of manifolds with non-smooth metrics. In particular, we will discuss some important results related to the Yamabe problem for closed manifolds and asymptotically Euclidean manifolds with rough metrics.