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The N-dimensional Hamiltonian H formed by a curved kinetic term (depending ona function f), a central potential (depending on a function U), a Diracmonopole term, and N centrifugal terms is shown to be quasi-maximallysuperintegrable for any choice of the functions f and U. This result is provenby making use of the underlying sl(2,R)-coalgebra symmetry of H in order toobtain a set of (2N-3) functionally independent integrals of the motion, thatare explicitly given. Such constants of the motion are "universal" since all ofthem are independent of both f and U. This Hamiltonian describes the motion ofa particle on any ND spherically symmetric curved space (whose metric isspecified by a function f) under the action of an arbitrary central potental U,and includes simultaneously a monopole-type contribution together with Ncentrifugal terms that break the spherical symmetry. Moreover, we show that twoappropriate choices for U provide the "intrinsic" oscillator and the KCpotentials on these curved manifolds. As a byproduct, the MIC-Kepler, theTaub-NUT and the so called multifold Kepler systems are shown to belong to thisclass of superintegrable Hamiltonians, and new generalizations thereof areobtained. The Kepler and oscillator potentials on N-dimensional generalizationsof the four Darboux surfaces are discussed as well.

Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles