  We suggest a generalization of the Feynman path integral to an integral over
random surfaces. The proposed action is proportional to the linear size of the
random surfaces and is called gonihedric. The convergence and the properties of
the partition function are analysed. The model can also be formulated as a spin
system with identical partition function. The spin system represents a
generalisation of the Ising model with ferromagnetic, antiferromagnetic and
quartic interactions. Higher symmetry of the model allows to construct dual
spin systems in three and four dimensions. In three dimensions the transfer
matrix describes the propagation of closed loops and we found its exact
spectrum. It is a unique exact solution of the tree-dimensional statistical
spin system. In three and four dimensions the system exhibits the second order
phase transitions. The gonihedric spin systems have exponentially degenerated
vacuum states separated by the potential barriers and can be used as a storage
of binary information.
