  Magnetic materials in which it is possible to control the topology of their
magnetic order in real space or the topology of their magnetic excitations in
reciprocal space are highly sought-after as platforms for alternative data
storage and computing architectures. Here we show that multiferroic insulators,
owing to their magneto-electric coupling, offer a natural and advantageous way
to address these two different topologies using laser fields. We demonstrate
that via a delicate balance between the energy injection from a high-frequency
laser and dissipation, single skyrmions -- archetypical topological magnetic
textures -- can be set into motion with a velocity and propagation direction
that can be tuned by the laser field amplitude and polarization, respectively.
Moreover, we uncover an ultrafast Floquet magnonic topological phase transition
in a laser-driven skyrmion crystal and we propose a new diagnostic tool to
reveal it using the magnonic thermal Hall conductivity.
