  The partition function, $U$, the number of available states in an atom or
molecules, is crucial for understanding the physical state of any astrophysical
system in thermodynamic equilibrium. There are surprisingly few {\em useful}
discussions of the partition function's numerical value. Textbooks often define
$U$; some give tables of representative values, while others do a deep dive
into the theory of a dense plasma. Most say that it depends on temperature,
atomic structure, density, and that it diverges, that is, it goes to infinity,
at high temperatures, but few give practical examples. We aim to rectify this.
We show that there are two limits, one and two-electron (or closed-shell)
systems like H or He, and species with a complicated electronic structure like
C, N, O, and Fe. The high-temperature divergence does not occur for one and
two-electron systems in practical situations since, at high temperatures,
species are collisionally ionized to higher ionization stages and are not
abundant. The partition function is then close to the statistical weight of the
ground state. There is no such simplification for many-electron species. $U$ is
temperature-sensitive across the range of temperatures where an ion is abundant
but remains finite at even the highest practical temperatures. The actual value
depends on highly uncertain truncation theories in high-density plasmas. We
show that there are various theories for continuum lowering but that they are
not in good agreement. This remains a long-standing unsolved problem.
