  Via Monte Carlo studies of the frustrated XY or classical planar model we
demonstrate the possibility of a finite (nonzero) temperature spin/gauge glass
phase in two dimensions. Examples of both periodic and quasiperiodic two
dimensional lattices, where a high temperature paramagnetic phase changes to a
spin/gauge glass phase with the lowering of temperature, are presented. The
existence of the spin/gauge glass phase is substantiated by our study of the
temperature dependence of the Edwards-Anderson order parameter, spin glass
susceptibility, linear susceptibility and the specific heat. Finite size
scaling analysis of spin glass susceptibility and order parameter yields a
nonzero critical temperature and exponents that are in close agreement with
those obtained by Bhatt and Young in their random ${\pm J}$ Ising model study
on a square lattice. These results suggest that certain periodic and
quasiperiodic two-dimensional arrays of superconducting grains in suitably
chosen transverse magnetic fields should behave as superconducting glasses at
low temperatures.
