(Joint with Peter Haine) Many geometric structures can be described as \emph{algebras} for a suitable monad. E.g., compact Hausdorff spaces are exactly the algebras for the ultrafilter monad on sets. I will explain this point of view through several concrete examples, starting with free abelian groups and rings, and then discussing ultrafilters, Eilenberg--Moore categories, and Kleisli categories. The emphasis will be on examples rather than category theory. The main application is in logarithmic geometry. I will explain how logarithmic maps can be realized as Kleisli morphisms for a natural monad built from Olsson's logarithmic stack. This leads to a functorial tropicalization construction for logarithmic schemes, recovering the usual fans of toric varieties and toroidal embeddings while remaining functorial for arbitrary logarithmic maps. (This talk is meant to be accessible to graduate students)