The Lojasiewicz-Simon gradient inequality is a generalization, due to Leon Simon (1983), to analytic or Morse-Bott functionals on Banach manifolds of the finite-dimensional gradient inequality, due to Stanislaw Lojasiewicz (1963), for analytic functions on Euclidean space. In this talk, we shall discuss several generalizations of the Lojasiewicz-Simon gradient inequality and a selection of their applications, including discreteness of the energy spectrum for harmonic maps from Riemann surfaces into analytic Riemannian manifolds. This is joint work with Manos Maridaks.