  In this paper we study the chord index of virtual knots, which can be thought
of as an extension of the chord parity. We show how to use the chord index to
define finite type invariants of virtual knots. The notions of indexed Jones
polynomial and indexed quandle are introduced, which generalize the classical
Jones polynomial and knot quandle respectively. Some applications of these new
invariants are discussed. We also study how to define a generalized chord index
via a fixed finite biquandle. Finally the chord index and its applications in
twisted knot theory are discussed.
