The utility of the concept of negative energy waves (NEW) in hydrodynamics is discussed. Examples are given of the excitation of waves by flow past elastic membranes, and of the amplification and generation of capillary-gravity and internal waves in liquids in the presence of vertically inhomogeneous flows. The concepts of ``linear'' and ``nonlinear'' energy are introduced, and it is shown that energy defined as the first integral of the equations of motion linearized against the flow background can be negative, whereas the inclusion of all the quadratic terms in the expression for the energy can give a positive value. Nonlinear processes associated with NEW are also discussed, as is the radiation instability of oscillators in hydrodynamics. The review is largely based on the authors' own work.