In its original form, the Atiyah-Singer Index Theorem equates two glob
al quantities of a closed manifold, one analytic (the index of an elli
ptic operator) and one topological (a characteristic number). Because
it relates invariants from different branches of mathematics, the Inde
x Theorem has many applications and extensions to differential geometr
y, K-theory, mathematical physics, and other fields. This report focus
es on advances in geometric aspects of index theory. For operators nat
urally associated to a Riemannian metric on a closed manifold, the top
ological side of the Index Theorem can often be expressed as the integ
ral of local (i.e. pointwise) curvature expression. We win first discu
ss these local refinements in 1, which arise naturally in heat equatio
n proofs of the Index Theorem. In 2,3, we discuss further developments
in index theory which lead to spectral invariants, the eta invariant
and the determinant of an elliptic operator, that are definitely nonlo
cal. Finally, in 4 we point out some recent connections among these no
nlocal invariants and classical index theory.