  A fully relational quantum theory necessarily requires an account of changes
of quantum reference frames, where quantum reference frames are quantum systems
relative to which other systems are described. By introducing a relational
formalism which identifies coordinate systems with elements of a symmetry group
$G$, we define a general operator for reversibly changing between quantum
reference frames associated to a group $G$. This generalises the known operator
for translations and boosts to arbitrary finite and locally compact groups,
including non-Abelian groups. We show under which conditions one can uniquely
assign coordinate choices to physical systems (to form reference frames) and
how to reversibly transform between them, providing transformations between
coordinate systems which are `in a superposition' of other coordinate systems.
We obtain the change of quantum reference frame from the principles of
relational physics and of coherent change of reference frame. We prove a
theorem stating that the change of quantum reference frame consistent with
these principles is unitary if and only if the reference systems carry the left
and right regular representations of $G$. We also define irreversible changes
of reference frame for classical and quantum systems in the case where the
symmetry group $G$ is a semi-direct product $G = N \rtimes P$ or a direct
product $G = N \times P$, providing multiple examples of both reversible and
irreversible changes of quantum reference system along the way. Finally, we
apply the relational formalism and changes of reference frame developed in this
work to the Wigner's friend scenario, finding similar conclusions to those in
relational quantum mechanics using an explicit change of reference frame as
opposed to indirect reasoning using measurement operators.
