  In recent years, progress toward the classification of superintegrable
systems with higher order integrals of motion has been made. In particular, a
complete classification of all exotic potentials with a third or a fourth order
integrals, and allowing separation of variables in Cartesian coordinates. All
doubly exotic potentials with a fifth order integral have also been completely
classified. It has been demonstrated how the Chazy class of third order
differential equations plays an important role in solving determining
equations. Moreover, taking advantage of various operator algebras defined as
Abelian, Heisenberg, Conformal and Ladder case of operator algebras, we
re-derived these models. These new techniques also provided further examples of
superintegrable Hamiltonian with integrals of arbitrary order. It has been
conjectured that all quantum superintegrable potentials that do not satisfy any
linear equation satisfy nonlinear equations having the Painlev\'e property. In
addition, it has been discovered that their integrals naturally generate
finitely generated polynomial algebras and the representations can be exploited
to calculate the energy spectrum. For certain very interesting cases associated
with exceptional orthogonal polynomials, these algebraic structures do not
allow to calculate the full spectrum and degeneracies. It has been demonstrated
that alternative sets of integrals which can be build and used to provide a
complete solution. This this allow to make another conjecture i.e. that higher
order superintegrable systems can be solved algebraically, they require
alternative set of integrals than the one provided by a direct approach.
