The notion of incremental stability was proposed by several researchers as a strong property of dynamical and control systems. Incremental stability describes the convergence of trajectories with respect to themselves, rather than with respect to an equilibrium point or a particular trajectory. Similarly to stability, Lyapunov functions play an important role in the study of incremental stability. In this paper, we propose new notions of incremental Lyapunov functions which are coordinate independent and provide the description of incremental stability in terms of the proposed Lyapunov functions. Moreover, we develop a backstepping design approach providing a recursive way of constructing controllers, enforcing incremental stability, as well as incremental Lyapunov functions. The effectiveness of the proposed method is illustrated by synthesizing a controller rendering a single-machine infinitebus electrical power system incrementally stable.