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Several important properties of positive semidefinite processes of Ornstein-Uhlenbeck type are analysed. It is shown that linear operators of the form X → AX + XA T with A ∈ M d (R) are the only ones that can be used in the definition provided one demands a natural non-degeneracy condition. Furthermore, we analyse the absolute continuity properties of the stationary distribution (especially when the driving matrix subordinator is the quadratic variation of a d-dimensional Lévy process) and study the question of how to choose the driving matrix subordinator in order to obtain a given stationary distribution. Finally, we present results on the first and second order moment structure of matrix subordinators, which is closely related to the moment structure of positive semidefinite Ornstein-Uhlenbeck type processes. The latter results are important for method of moments based estimation.Keywords: completely positive matrix; matrix subordinator; normal mixture; operator self-decomposable distributions; positive semidefinite Ornstein-Uhlenbeck type process; quadratic variation; second order structure; stationary distribution with A being a d × d-matrix, L a matrix subordinator (see [3]) and the initial value Σ 0 a positive semidefinite matrix. Matrix subordinators are a generalisation of the one-This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2009, Vol. 15, No. 3, 754-773. This reprint differs from the original in pagination and typographic detail.