  This Topical Review describes the multitude of unconventional behaviors in
the hidden order, heavy fermion, antiferromagnetic and superconducting phases
of the intermetallic compound URu$_2$Si$_2$ when tuned with pressure, magnetic
field, and substitutions for all three elements. Such `perturbations' result in
a variety of new phases beyond the mysterious hidden order that are only now
being slowly understood through a series of state-of-the-science
experimentation, along with an array of novel theoretical approaches. Despite
all these efforts spanning more than 30 years, hidden order (HO) remains
puzzling and non-clarified, and the search continues in 2019 into a fourth
decade for its final resolution. Here we attempt to update the present
situation of URu$_2$Si$_2$ importing the latest experimental results and
theoretical proposals. First, let us consider the pristine compound as a
function of temperature and report the recent measurements and models relating
to its heavy Fermi liquid crossover, its HO and superconductivity (SC). Recent
experiments and theories are surmized that address four-fold symmetry breaking
(or nematicity), Isingness and unconventional excitation modes. Second, we
review the pressure dependence of URu$_2$Si$_2$ and its transformation to
antiferromagnetic long-range order. Next we confront the dramatic high
magnetic-field phases requiring fields above 40 T. And finally, we attempt to
answer how does random substitutions of other $5f$ elements for U, and $3d$,
$4d$, and $5d$ elements for Ru, and even P for Si affect and transform the HO.
Commensurately, recent theoretical models are summarized and then related to
the intriguing experimental behavior.
