This work is devoted to investigation of tunnelling and switching phenomena arising
in Condensed Matter Physics. In the first part I investigate quantum tunnelling of
a single Abrikosov vortex in a superconducting quantum dot. The escape of a vortex
carrying magnetic flux leads to a switching of a quantum dot to non-magnetic state. Tunnelling of a vortex manifests itself as a spontaneous relaxation of magnetization. I
showed that it is possible to use an instanton technique based on path integral formalism
even in presence of forces breaking time-reversal symmetry. [Continues.
