  We present a pedagogical overview of recent theoretical work on
unconventional quantum phases and quantum phase transitions in condensed matter
systems. Strong correlations between electrons can lead to a breakdown of two
traditional paradigms of solid state physics: Landau's theories of Fermi
liquids and phase transitions. We discuss two resulting "exotic" states of
matter: topological and critical spin liquids. These two quantum phases do not
display any long-range order even at zero temperature. In each case, we show
how a gauge theory description is useful to describe the new concepts of
topological order, fractionalization and deconfinement of excitations which can
be present in such spin liquids. We make brief connections, when possible, to
experiments in which the corresponding physics can be probed. Finally, we
review recent work on deconfined quantum critical points. The tone of these
lecture notes is expository: focus is on gaining a physical picture and
understanding, with technical details kept to a minimum.
