  I begin with a proposed global phase diagram of the cuprate superconductors
as a function of carrier concentration, magnetic field, and temperature, and
highlight its connection to numerous recent experiments. The phase diagram is
then used as a point of departure for a pedagogical review of various quantum
phases and phase transitions of insulators, superconductors, and metals. The
bond operator method is used to describe the transition of dimerized
antiferromagnetic insulators between magnetically ordered states and spin-gap
states. The Schwinger boson method is applied to frustrated square lattice
antiferromagnets: phase diagrams containing collinear and spirally ordered
magnetic states, Z_2 spin liquids, and valence bond solids are presented, and
described by an effective gauge theory of spinons. Insights from these theories
of insulators are then applied to a variety of symmetry breaking transitions in
d-wave superconductors. The latter systems also contain fermionic
quasiparticles with a massless Dirac spectrum, and their influence on the order
parameter fluctuations and quantum criticality is carefully discussed. I
conclude with an introduction to strong coupling problems associated with
symmetry breaking transitions in two-dimensional metals, where the order
parameter fluctuations couple to a gapless line of fermionic excitations along
the Fermi surface.
