This book contains the introduction and preliminaries of Lie and Noether theory. The conservation laws of the (1 + n)-dimensional heat equation on curved surfaces and the derivation of conserved quantities of the (1 + 2)-dimensional wave equation on different surfaces is discussed in detail. This book deals with conserved quantities of the (1 + n)-dimensional heat equation. The effect of a background metric on the symmetries of the nonlinear (1 + 2)-dimensional heat equation is also presented here, then all possible reduced equations up to conjugacy classes of the time dependent Ginzburg-Landau model are calculated and then the (1 + 2)-dimensional heat equation on the torus is considered, subalgebras are classified up to conjugacy classes and finally some solutions are calculated. A complete group classification of the (1 + n)-dimensional Klein-Gordon equation and nonlinear (1 + 2)-dimensional wave equation on the sphere and torus is presented here.