  We give a pedagogical introduction to the nonassociative structures arising
from recent developments in quantum mechanics with magnetic monopoles, in
string theory and M-theory with non-geometric fluxes, and in M-theory with
non-geometric Kaluza-Klein monopoles. After a brief overview of the main
historical appearences of nonassociativity in quantum mechanics, string theory
and M-theory, we provide a detailed account of the classical and quantum
dynamics of electric charges in the backgrounds of various distributions of
magnetic charge. We apply Born reciprocity to map this system to the phase
space of closed strings propagating in R-flux backgrounds of string theory, and
then describe the lift to the phase space of M2-branes in R-flux backgrounds of
M-theory. Applying Born reciprocity maps this M-theory configuration to the
phase space of M-waves probing a non-geometric Kaluza-Klein monopole
background. These four perspective systems are unified by a covariant 3-algebra
structure on the M-theory phase space.
