Abstract: Tautological systems are vast generalizations of $A$-hypergeometric systems to the case of an arbitrary reductive algebraic group. Much of the interest in such systems has come from their application to period integrals of Calabi-Yau hypersurfaces. As with $A$-hypergeometric systems, part of the input data is a parameter $\beta$. I will discuss joint work with P. Gorlach, T. Reichelt, C. Sevenheck, and U. Walther discussing criteria which bounds the number of parameters $\beta$ which give a non-trivial tautological system.