This is a series of four lectures presented at the 2015 Enrico Fermi Summer School in Varenna. The aim of the lectures is to give an introduction to the hydrodynamics of active matter concentrating on low-Reynolds-number examples such as cells and molecular motors. Lecture 1 introduces the hydrodynamics of single active particles, covering the Stokes equation and the Scallop Theorem, and stressing the link between autonomous activity and the dipolar symmetry of the far flow field. In lecture 2 I discuss applications of this mathematics to the behaviour of microswimmers at surfaces and in external flows, and describe our current understanding of how swimmers stir the surrounding fluid. Lecture 3 concentrates on the collective behaviour of active particles, modelled as an active nematic. I write down the equations of motion and motivate the form of the active stress. The resulting hydrodynamic instability leads to a state termed “active turbulence” characterised by strong jets and vortices in the flow field and the continual creation and annihilation of pairs of topological defects. Lecture 4 compares simulations of active turbulence to experiments on suspensions of microtubules and molecular motors. I introduce lyotropic active nematics and discuss active anchoring at interfaces.