  We develop a quantum theory of magnetic skyrmions and antiskyrmions in a
spin-1/2 Heisenberg magnet with frustrating next-nearest neighbor interactions.
Using exact diagonalization we show numerically that a quantum skyrmion exists
as a stable many-magnon bound state and investigate its quantum numbers. We
then derive a phenomenological Schr\"odinger equation for the quantum skyrmion
and its internal degrees of freedom. We find that quantum skyrmions have highly
unusual properties. Their bandwidth is exponentially small and arises from
tunneling processes between skyrmion and antiskyrmion. The bandstructure
changes both qualitatively and quantitatively when a single spin is added or
removed from the quantum skyrmion, reflecting a locking of angular momentum and
spin quantum numbers characteristic for skyrmions. Additionally, while for weak
forces the quantum skyrmion is accelerated parallel to the force, it moves in a
perpendicular direction for stronger fields.
