B-series are a special form of Taylor series, where the terms are indexed by rooted trees. The theory originated by the seminal work on numerical integration by John Butcher half a century ago. The theory has developed to become arguably the most important tool in the study of structure preservation of numerical integrators. Geometrically B-series are intimately connected to the geometry of Euclidean spaces. Lie-Butcher series is a generalisation which combines B-series with Lie series, aimed at the study of flows on manifolds. In this talk we will discuss the interplay between the algebraic and geometric structures underlying Lie-Butcher series. We will review various recent results and work in progress, both for ODEs and in the theory of Rough Paths (stochastic DEs).