  To illustrate the campaign to understand heavy quarkonium systems, I focus on
a puzzling new state, $X(3872) \to \pi^+\pi^- J/\psi$. Studying the influence
of open-charm channels on charmonium properties leads us to propose a new
charmonium spectroscopy: additional discrete charmonium levels that can be
discovered as narrow resonances of charmed and anticharmed mesons. I recall
some expectations for a new spectroscopy of mesons with beauty and charm, the
$B_c$ ($b\bar{c}$) system. Throughout, I call attention to open issues for
theory and experiment.
