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.
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