I will introduce the basics of toroidal algebras and certain extensions of them, together with their representations using vertex operator algebraic techniques. I will also introduce certain extensions of toroidal algebras including a generalization to lie super algebras. The purpose of this generalization will be related to the other main topic of the lecture, which will be the equivariant cohomology of moduli spaces of sheaves on elliptic surfaces which serve as weight spaces for representations of the algebras. The action of autoequivalences of the derived category of the elliptic surfaces will be identified with automorphisms of toroidal algebras. Time permitting, we will discuss other representations of toroidal lie algebras related to flags of sheaves and their relations with superconformal algebras.