  The history of modern condensed matter physics may be regarded as the
competition and reconciliation between Stoner's and Anderson's physical
pictures, where the former is based on momentum-space descriptions focusing on
long wave-length fluctuations while the latter is based on real-space physics
emphasizing emergent localized excitations. In particular, these two view
points compete with each other in various nonperturbative phenomena, which
range from the problem of high T$_{c}$ superconductivity, quantum spin liquids
in organic materials and frustrated spin systems, heavy-fermion quantum
criticality, metal-insulator transitions in correlated electron systems such as
doped silicons and two-dimensional electron systems, the fractional quantum
Hall effect, to the recently discussed Fe-based superconductors. An approach to
reconcile these competing frameworks is to introduce topologically nontrivial
excitations into the Stoner's description, which appear to be localized in
either space or time and sometimes both, where scattering between itinerant
electrons and topological excitations such as skyrmions, vortices, various
forms of instantons, emergent magnetic monopoles, and etc. may catch
nonperturbative local physics beyond the Stoner's paradigm. In this review
article we discuss nonperturbative effects of topological excitations on
dynamics of correlated electrons. ......
