  Diffusion approximations have been a popular tool for performance analysis in
queueing theory, with the main reason being tractability and computational
efficiency. This dissertation is concerned with establishing theoretical
guarantees on the performance of steady-state diffusion approximations of
queueing systems. We develop a modular framework based on Stein's method that
allows us to establish error bounds, or convergence rates, for the
approximations. We apply this framework three queueing systems: the Erlang-C,
Erlang-A, and $M/Ph/n+M$ systems.
  The former two systems are simpler and allow us to showcase the full
potential of the framework. Namely, we prove that both Wasserstein and
Kolmogorov distances between the stationary distribution of a normalized
customer count process, and that of an appropriately defined diffusion process
decrease at a rate of $1/\sqrt{R}$, where $R$ is the offered load. Futhermore,
these error bounds are \emph{universal}, valid in any load condition from
lightly loaded to heavily loaded. For the Erlang-C model, we also show that a
diffusion approximation with state-dependent diffusion coefficient can achieve
a rate of convergence of $1/R$, which is an order of magnitude faster when
compared to approximations with constant diffusion coefficients.
