  In this paper we review important aspects of semiparametric theory and
empirical processes that arise in causal inference problems. We begin with a
brief introduction to the general problem of causal inference, and go on to
discuss estimation and inference for causal effects under semiparametric
models, which allow parts of the data-generating process to be unrestricted if
they are not of particular interest (i.e., nuisance functions). These models
are very useful in causal problems because the outcome process is often complex
and difficult to model, and there may only be information available about the
treatment process (at best). Semiparametric theory gives a framework for
benchmarking efficiency and constructing estimators in such settings. In the
second part of the paper we discuss empirical process theory, which provides
powerful tools for understanding the asymptotic behavior of semiparametric
estimators that depend on flexible nonparametric estimators of nuisance
functions. These tools are crucial for incorporating machine learning and other
modern methods into causal inference analyses. We conclude by examining related
extensions and future directions for work in semiparametric causal inference.
