This talk is the first a two-part sequence of talks in which we will discuss classical and more recent results in the theory nonlinear wave equations. In particular, we will consider small-data solutions to equations of the form $(partial_t^2 - Delta)u = Q(u,u',u'')$ where $Q$ is the nonlinearity which is quadratic in $u$ and its derivatives and linear in $u''$. We will mainly cover the invariant vector field method of Klainerman and how how it has been employed in proving local and long-time existence results for these kinds of equations. The main goal of this talk is to provide sufficient background for the second talk, which will discuss nonlinear wave equations where the spatial variable $x$ lies in an exterior domain $mathbb{R}^3 backslash mathcal{K}$, where $mathcal{K}$ is a smooth, bounded domain.