Mesophases are intermediate between unstructured fluids and fully ordered crystalline solids. They often self-assemble at the mesoscale, albeit into defected microstructures. We describe research on a mesoscopic description of the response of these phases to shears, including elastic and dissipative contributions. Particular applications to lamellar, hexagonal, and gyroid phases are considered. We also use this description to analyze the motion of extended defects such as grain boundaries, and address the issue of orientation selection under imposed oscillatory shears.