Abstract: We will discuss the local cohomology modules supported at the nullcone ideal for the classical actions of the linear algebraic groups. We calculate the arithmetic rank of the nullcone ideals in every characteristic and establish sharp vanishing results for local cohomology modules supported at these ideals over any commutative Notherian ring which mirror the corresponding results for determinantal ideals. This is joint work with Jack Jeffries, Anurag Singh, and Uli Walther.