Infinitely divisible distributions and processes have been the object of extensive research not only from the theoretical point of view but also for practical use, for example, in queueing theory or mathematical finance. In this thesis, we will study some of their subclasses with a view towards financial modeling. As generalizations of stable distributions, we study the tempered stable distributions and introduce the new classes of layered stable distributions as well as the mixed stable distributions, along with the corresponding Levy processes. As a further generalization of infinitely divisible processes, fractional tempered stable motions are defined. These theoretical studies will be complemented by some more practical ones, such as the simulation of sample paths, parameter estimations, financial portfolio hedging, and solving stochastic differential equations.