We study the ground state properties of classical Coulomb charges interacting with a 1∕r potential moving on a plane but confined either by a circular hard-wall boundary or by a harmonic potential. The charge density in the continuum limit is determined analytically and is nonuniform. Because of the nonuniform density there are both disclinations and dislocations present and their distribution across the system is calculated and shown to be in agreement with numerical studies of the ground state (or at least low-energy states) of N charges, where values of N up to 5000 have been studied. A consequence of these defects is that although the charges locally form into a triangular lattice structure, the lattice lines acquire a marked curvature. A study is made of conformal crystals to illuminate the origin of this curvature. The scaling of various terms which contribute to the overall energy of the system of charges, viz., the continuum electrostatic energy, correlation energy, surface energy (and so on) as a function of the number of particles N is determined. “Magic number” clusters are those at special values of N whose energies take them below the energy estimated from the scaling forms and are identified with charge arrangements of high symmetry.