  The paper is devoted to the index theory of orbital and transverse elliptic
operators on manifolds with a proper Lie group action. It corrects errors of my
previous paper (published in JNCG in 2016) on transverse operators and contains
new results. The two index theories, orbital and transverse, are very much
intertwined and interdependent, and are treated together. The theory of orbital
operators is developed from the basic definitions to the final index theorem.
The KK-theoretic proofs of index theorems for elliptic, t-elliptic and orbital
elliptic operators are given in sections 9, 10, 11. Throughout the paper, we
use a simpler method in constructing pseudo-differential operators.
