  We propose a scheme realizing topological insulators and quantized pumps for
magnon excitations, based on strongly-interacting two-component ultracold atoms
trapped in optical superlattices. Specifically, we show how to engineer the
Su-Schrieffer-Heeger model for magnons using state-independent superlattices,
and the Rice-Mele model using state-dependent superlattices. We describe
realistic experimental protocols to detect the topological signatures of magnon
excitations in these two models. In particular, we show that the
non-equilibrium dynamics of a single magnon can be exploited to directly detect
topological winding numbers and phase transitions. We also describe how
topological (quantized) pumps can be realized with magnons, and study how this
phenomenon depends on the initial magnon state preparation. Our study opens a
new avenue for exploring magnonic topological phases of matter and their
potential applications in the context of topological magnon transport.
