  Continuous global symmetries are expected to be broken by gravity, which can
lead to important phenomenological consequences. A prime example is the threat
that this poses to the viability of the Peccei-Quinn solution to the strong CP
problem. In this paper, we explore the impact of wormholes as a source of
global symmetry breaking by gravity. We review the current status of wormholes
and global symmetries and note that, surprisingly, the axion has a quality
problem within non-perturbative Einstein gravity. Although these wormholes lead
to a large breaking of global symmetries, we show that their effect is
nonetheless relevant for the model building of gauge protected axions. We also
find wormhole solutions within two scenarios: (i) an extended global symmetry
group within Einstein gravity, and (ii) U(1) wormholes within the low-energy
limit of an open String Theory. The former allows us to show that the concept
of a global symmetry in General Relativity is somewhat ill-defined. The latter
illustrates that for motivated values of the string coupling constant, axions
appear to have a quality problem within the open String Theory we consider.
