  The biological fluids encountered by self-propelled cells display complex
microstructures and rheology. We consider here the general problem of
low-Reynolds number locomotion in a complex fluid. {Building on classical work
on the transport of particles in viscoelastic fluids,} we demonstrate how to
mathematically derive three integral theorems relating the arbitrary motion of
an isolated organism to its swimming kinematics {in a non-Newtonian fluid}.
These theorems correspond to three situations of interest, namely (1) squirming
motion in a linear viscoelastic fluid, (2) arbitrary surface deformation in a
weakly non-Newtonian fluid, and (3) small-amplitude deformation in an
arbitrarily non-Newtonian fluid. Our final results, valid for a wide-class of
{swimmer geometry,} surface kinematics and constitutive models, at most require
mathematical knowledge of a series of Newtonian flow problems, and will be
useful to quantity the locomotion of biological and synthetic swimmers in
complex environments.
