In this talk, we describe parametrized topological complexity introduced by Cohen, Farber and Weinberger. Motivated by practical considerations in motion planning, we investigate which symmetry constraints should naturally be imposed on parametrized motion planning problems. This leads us to study several symmetry-adapted variants of parametrized topological complexity, including the equivariant, invariant, symmetrized, monoidal, and monoidal symmetrized versions.
