  Many systems in nature and the synthetic world involve ordered arrangements
of units on two-dimensional surfaces. We review here the fundamental role payed
by both the topology of the underlying surface and its detailed curvature.
Topology dictates certain broad features of the defect structure of the ground
state but curvature-driven energetics controls the detailed structured of
ordered phases. Among the surprises are the appearance in the ground state of
structures that would normally be thermal excitations and thus prohibited at
zero temperature. Examples include excess dislocations in the form of grain
boundary scars for spherical crystals above a minimal system size, dislocation
unbinding for toroidal hexatics, interstitial fractionalization in spherical
crystals and the appearance of well-separated disclinations for toroidal
crystals. Much of the analysis leads to universal predictions that do not
depend on the details of the microscopic interactions that lead to order in the
first place. These predictions are subject to test by the many experimental
soft and hard matter systems that lead to curved ordered structures such as
colloidal particles self-assembling on droplets of one liquid in a second
liquid. The defects themselves may be functionalized to create ligands with
directional bonding. Thus nano to meso scale superatoms may be designed with
specific valency for use in building supermolecules and novel bulk materials.
Parameters such as particle number, geometrical aspect ratios and anisotropy of
elastic moduli permit the tuning of the precise architecture of the superatoms
and associated supermolecules. Thus the field has tremendous potential from
both a fundamental and materials science/supramolecular chemistry viewpoint.
