This book is concerned with the development of an adaptive numerical algorithm for solving parabolic PDEs in Cartesian geometry in one, two and three space dimensions and the application to thermodiffusive flame instabilities. Applying a finite volume discretization with explicit time integration, both of second order, we employ an adaptive multiresolution scheme to represent the solution on locally refined nested grids. We validate this method by solving several test problems in one, two and three space dimensions, like convection-diffusion and viscous Burgers equations, in order to show its accuracy and its efficiency. We then apply the adaptive method to study the stability behavior of premixed flames in the thermo-diffusive approximation. We determine the limit between stable and pulsating planar flames. We also perform numerical simulations of flame ball interactions with adiabatic walls. The last result concerns the interaction of a flame ball with a vortex.