In the present thesis, we consider the evolution equation (Cauchy problem) which is the basis for our study. We show how various linear partial differential equations can be transformed into the Cauchy problem form. Solving the Cauchy problem is equivalent to find a family of evolution operators T(t) which sends the initial state of the system to the solution state at a later time t. It turns out that this family of operators T(t) must satisfy some properties which we call semigroup properties. We state the Hille-Yosida and Lumer-Phillips theorems to characterize contraction semigroups. Moreover, we apply these theorems to the heat and wave equations as examples. We also consider strongly continuous operator groups and Stone's theorem. Finally, we give some essential conditions to obtain wellposed evaluation equation and introduce an inhomogeneous Cauchy problem.