  The survival of many microorganisms, like \textit{Leptospira} or
\textit{Spiroplasma} bacteria, can depend on their ability to navigate towards
regions of favorable viscosity. While this ability, called viscotaxis, has been
observed in several bacterial experiments, the underlying mechanism remains
unclear. Here, we provide a framework to study viscotaxis of self-propelled
swimmers in slowly varying viscosity fields and show that suitable body shapes
create viscotaxis based on a systematic asymmetry of viscous forces acting on a
microswimmer. Our results shed new light on viscotaxis in \textit{Spiroplasma}
and \textit{Leptospira} and suggest that dynamic body shape changes exhibited
by both types of microorganisms may have an unrecognized functionality: to
prevent them from drifting to low viscosity regions where they swim poorly. The
present theory classifies microswimmers regarding their ability to show
viscotaxis and can be used to design synthetic viscotactic swimmers, e.g.\ for
delivering drugs to a target region distinguished by viscosity.
