One of the main developments in optimization over the last 20 years is Semi-Definite Programming. It treats problems which can be expressed as a Linear Matrix Inequality (LMI). Any such problem is necessarily convex, so the determining the scope and range of applicability comes down to the question: How much more restricted are LMIs than Convex Matrix Inequalities? The talk gives a survey of what is known on this issue and will be accessible to about anybody. There are several main branches of this pursuit. First there are two fundamentally different classes of linear systems problems. Ones whose statements do depend on the dimension of the system "explicitly" and ones whose statements do not. Dimension dependent systems problems lead to traditional semialgebraic geometry problems, while dimension free systems problems lead directly to problems in matrix unknowns and a new area which might be called noncommutative semialgebraic geometry. Most classic problems of control lead to noncommutative problems. In this talk after laying out the distinctions above we give results and conjectures on the answer to the LMI vs convexity question.