  We introduce a shifted analog of the plactic monoid of Lascoux and
Sch\"utzenberger, the \emph{shifted plactic monoid}. It can be defined in two
different ways: via the \emph{shifted Knuth relations}, or using Haiman's mixed
insertion.
  Applications include: a new combinatorial derivation (and a new version of)
the shifted Littlewood-Richardson Rule; similar results for the coefficients in
the Schur expansion of a Schur $P$-function; a shifted counterpart of the
Lascoux-Sch\"utzenberger theory of noncommutative Schur functions in plactic
variables; a characterization of shifted tableau words; and more.
