  If history teaches us anything, it is that the next breakthrough in
superconductivity will not be the result of surveying the history of past
breakthroughs, as they have almost always been a matter of serendipity
resulting from undirected exploration into new materials. Still, there is
reason to reflect on recent advances, work toward higher T_c of even an
incremental nature, and recognize that it is important to explore avenues
currently believed to be unpromising even as we attempt to be rational. In this
paper we look at two remarkable new unusually high temperature superconductors
(UHTS), MgB2 with Tc=40 K and (in less detail) high pressure Li with Tc=20 K,
with the aim of reducing their unexpected achievements to a simple and clear
understanding. We also consider briefly other UHTS systems that provide still
unresolved puzzles; these materials include mostly layered structures, and
several with strongly bonded C-C or B-C substructures. What may be possible in
phonon-coupled superconductivity is reconsidered in the light of the
discussion.
