  The virial expansion provides a non-perturbative view into the thermodynamics
of quantum many-body systems in dilute regimes. While powerful, the expansion
is challenging as calculating its coefficients at each order $n$ requires
analyzing (if not solving) the quantum $n$-body problem. In this work, we
present a comprehensive review of automated algebra methods, which we developed
to calculate high-order virial coefficients. The methods are computational but
non-stochastic, thus avoiding statistical effects; they are also for the most
part analytic, not numerical, and amenable to massively parallel computer
architectures. We show formalism and results for coefficients characterizing
the thermodynamics (pressure, density, energy, static susceptibilities) of
homogeneous and harmonically trapped systems, and explain how to generalize
them to other observables such as the momentum distribution, Tan's contact, and
the structure factor.
