The book is organized as follows. We concentrate our attention on three subjects of semigroup theory: characterization, spectral theory and asymptotic behavior. By characterization , we understand the problem of describing special properties of a semigroup, such as positivity, through the generator. By spectral theory we mean the investigation of the spectrum of a generator. Asymptotic behavior refers to the orbits of the initial values under a given semigroup and phenomena such as stability.