A quasi-infinitely divisible distribution on $mathbb{R}^d$ is a probability distribution $mu$ on $mathbb{R}^d$ whose characteristic function can be written as the quotient of the characteristic functions of two infinitely divisible distributions on $mathbb{R}^d$. Equivalently, it can be characterised as a probability distribution whose characteristic function has a Levy--Khintchine type representation with a signed Levy measure, a so called quasi--Levy measure, rather than a Levy measure. A systematic study of such distributions in the univariate case has been carried out in Lindner, Pan and Sato cite{lindner}. The goal of the present paper is to collect some known results on multivariate quasi-infinitely divisible distributions and to extend some of the univariate results to the multivariate setting. In particular, conditions for weak convergence, moment and support properties are considered. A special emphasis is put on examples of such distributions and in particular on $mathbb{Z}^d$-valued quasi-infinitely divisible distributions.
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