  We introduce and study knotoids. Knotoids are represented by diagrams in a
surface which differ from the usual knot diagrams in that the underlying curve
is a segment rather than a circle. Knotoid diagrams are considered up to
Reidemeister moves applied away from the endpoints of the underlying segment.
We show that knotoids in $S^2$ generalize knots in $S^3$ and study the
semigroup of knotoids. We also discuss applications to knots and invariants of
knotoids.
