Complex fluids often show visco-elastic properties, i.e. a mix of solid- and liquid-like behavior. These materials are structured on a mesoscopic scale in contrast to the atomic structure of simple liquids. While being processed or used, they undergo moderate flows which can easily affect their mesostructure. In turn, structural modications feed back on the flow itself, leading to various nonlinear rheological responses. I will show that these structural modications can also trigger instabilities that develop on top of the nonlinear rheological behaviors. Those instabilities occur at flow rates where inertial effects are usually negligible. They are driven by elastic forces and give rise to complex spatio-temporal behavior and even disordered flows sometimes reminiscent of inertial turbulence. I will discuss various examples in living polymers and regular polymer systems flowing in different configurations : the benchmark Taylor-Couette flow and a 2D contraction flow that forms a liquid curtain.