A matrix map F (x) is said to be (matricially) convex, if u T F (x)u is a convex function for every u. In this paper, semideenite systems of the type F (x) 0, where F (x) is ma-tricially convex, are considered. This class of problems generalizes both aane semideenite inequalities as well as ordinary convex inequality systems. After establishing characterizations and properties of matricial convexity, a Newton-like algorithm is developed for the semideenite inequality problem. The global convergence of the method is proven and numerical experience is reported.