Abstract: Given the action of a compact Lie group on a smooth manifold, it is possible to define a class of (pseudo-)differential operators known as the transversally elliptic operators. The principal symbol of such an operator is only guaranteed to be invertible for nonzero covectors that are conormal to the group orbits. I will give the precise definition of these operators, and explain why they have a well-defined equivariant index, in spite of the fact that in general, they are not Fredholm operators.