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In this thesis, we investigate the class of quasi-infinitely divisible distributions. By definition, a distribution is quasi-infinitely divisible if its characteristic function can be written as the quotient of the characteristic functions of two infinitely divisible distributions. Hence, quasi-infinitely divisible distributions generalize the class of infinitely divisible distributions, which corresponds to the class of L\'evy processes in a natural way. Several distributional properties of infinitely divisible distributions can be characterized in terms of their characteristic triplet. In case of quasi-infinitely divisible distributions, this is more difficult, as was observed by Lindner, Pan and Sato (2018). There, Lindner et al. studied quasi-infinitely divisible distributions on the real line $\bR$ systematically. In Chapter 2 we focus on multivariate quasi-infinitely divisible distributions and collect various results on those. We derive some examples and study distributional properties of quasi-infinitely divisible distributions on $\bR^d$ for $d \in \bN$. In particular, we study their absolute continuity, weak convergence, support properties and the existence of certain moments for those distributions. Moreover, we study some topological properties of the class of quasi-infinitely divisible distributions on $\bR^d$. The class of quasi-infinitely divisible distributions on $\bR$ is dense in the class of all distributions on $\bR$ with respect to weak convergence. That this is no longer true in higher dimensions is shown in Chapter 3, where for $d \geq 2$ we give an example of a probability distribution on $\bR^d$, which cannot be approximated arbitrarily well by quasi-infinitely divisible distributions. In particular, it is shown that its characteristic function cannot be approximated arbitrarily well by a zero-free continuous function with respect to compact uniform convergence, and hence especially not by the characteristic function of a quasi-infinitely divisible distribution, since those are ...