Homotopy algebra is playing an increasing role in mathematical physics. Especially in the Hamiltonian and Lagrangian settings, it is intimately related to some of Alan's interests, e.g. Courant and Lie algebroids. There is a comparatively long history of such structure in cohomological physics in terms of equations that hold mod exact terms (typically, divergences) or only `on shell', meaning modulo the Euler-Lagrange equations of `motion'; more recently, higher homotopies have come into prominence. This talk is planned as an introduction to the basic point of view with a variety of applications to substantiate its relevance.