  The general description of superintegrable systems with one polynomial
integral of order $N$ in the momenta is presented for a Hamiltonian system in
two-dimensional Euclidean plane. We consider classical and quantum Hamiltonian
systems allowing separation of variables in polar coordinates. The potentials
can be classified into two major classes and their main properties are
described. We conjecture that a new infinite family of superintegrable
potentials in terms of the sixth Painlev\'e transcendent $P_6$ exists and
demonstrate this for the first few cases.
