This monograph presents new theory and methods of solving inverse Stefan problems for quasilinear parabolic equations in domains with free boundaries. This new approach to the theory of ill-posed problems is useful for the modelling of nonlinear processes with phase transforms in thermophysics and mechanics of continuous media. Regularization methods and algorithms are developed for the numerical solution of inverse Stefan problems ensuring substantial savings in computational costs. Results of calculations for important applications in a continuous casting and for the treatment of materials using laser technology are also given. This text should be of interest to students and researchers whose work involves partial differential equations, numerical analysis, phase transformation, mathematical modelling, industrial mathematics and the mathematics of physics.