  Superintegrable systems with monopole interactions in flat and curved spaces
have attracted much attention. For example, models in spaces with a Taub-NUT
metric are well-known to admit the Kepler-type symmetries and provide
non-trivial generalizations of the usual Kepler problems. In this paper, we
overview new families of superintegrable Kepler, MIC-harmonic oscillator and
deformed Kepler systems interacting with Yang-Coulomb monopoles in the flat and
curved Taub-NUT spaces. We present their higher-order, algebraically
independent integrals of motion via the direct and constructive approaches
which prove the superintegrability of the models. The integrals form symmetry
polynomial algebras of the systems with structure constants involving Casimir
operators of certain Lie algebras. Such algebraic approaches provide a deeper
understanding to the degeneracies of the energy spectra and connection between
wave functions and differential equations and geometry.
