Moving or free boundary problems (so-called Stefan problems) abound in many applications of diverse fields ranging from geophysical applications of global scale to crystal formation of molecular scale. The theory of Lagrange-Burmann expansions has been developed to cope with the imbedded nonlinear boundaries and boundary values. We review some recent development achieved in the Lagrange-Burmann expansions and show how the method can be applied to diverse Stefan solutions including gravity current problems, a typical free boundary problem. Unlike most of classical perturbation problems where higher order functions are almost always linearized, we are faced, in the free boundary problem, with integro-differential equations at higher order functions. Compared with the classical series solutions, the new solutions demonstrate markedly improved convergence having validity often over the entire physical time domain.