  Classical and quantum superintegrable systems have a long history and they
possess more integrals of motion than degrees of freedom. They have many
attractive properties, wide applications in modern physics and connection to
many domains in pure and applied mathematics. We overview two new families of
superintegrable Kepler-Coulomb systems with non-central terms and
superintegrable Hamiltonians with double singular oscillators of type $(n,N-n)$
in $N$-dimensional Euclidean space. We present their quadratic and polynomial
algebras involving Casimir operators of $so(N+1)$ Lie algebras that exhibit
very interesting decompositions $Q(3)\oplus so(N-1)$, $Q(3)\oplus so(n)\oplus
so(N-n)$ and the cubic Casimir operators. The realization of these algebras in
terms of deformed oscillator enables the determination of a finite dimensional
unitary representation. We present algebraic derivations of the degenerate
energy spectra of these systems and relate them with the physical spectra
obtained from the separation of variables.
