This chapter explores solid-state phase transformations, where crystals undergo rearrangement from one structure to another due to thermal or mechanical influences. Crystallography plays a key role in understanding these transformations, as it helps explain how new phases form within the solid state. The chapter presents analytical solutions for one-dimensional problems involving such transformations, focusing on moving grain boundaries. External control involves heating samples below the solidus temperature on a phase diagram, followed by controlled cooling. The transformation mechanism can be either diffusion-driven, where the concentration gradient ( \(\partial C/\partial x\) ) induces nucleation or precipitation of new phases, or diffusionless (displacive), where rapid cooling quenches the material, potentially leading to an amorphous structure. Diffusional processes, like annealing, involve slow cooling and resemble a random walk on the atomic level. In contrast, displacive transformations, achieved through rapid quenching, are more abrupt. Solid-state transformations involves mechanisms like nucleation and growth processes. Analytical descriptions of these processes are presented, allowing for comparisons with experimental observations. Ultimately, the goal is to optimize the control of these atomic rearrangements to achieve desired microstructures in materials.