  The classical random walk isomorphism theorems relate the local times of a
continuous-time random walk to the square of a Gaussian free field. A Gaussian
free field is a spin system that takes values in Euclidean space, and this
article generalises the classical isomorphism theorems to spin systems taking
values in hyperbolic and spherical geometries. The corresponding random walks
are no longer Markovian: they are the vertex-reinforced and vertex-diminished
jump processes. We also investigate supersymmetric versions of these formulas.
  Our proofs are based on exploiting the continuous symmetries of the
corresponding spin systems. The classical isomorphism theorems use the
translation symmetry of Euclidean space, while in hyperbolic and spherical
geometries the relevant symmetries are Lorentz boosts and rotations,
respectively. These very short proofs are new even in the Euclidean case.
  Isomorphism theorems are useful tools, and to illustrate this we present
several applications. These include simple proofs of exponential decay for spin
system correlations, exact formulas for the resolvents of the joint processes
of random walks together with their local times, and a new derivation of the
Sabot--Tarr\`es formula for the limiting local time of the vertex-reinforced
jump process.
