A regular foliation of a compact manifold is a partition of the manifold by connected immersed submanifolds of the same dimension, each of which is called a leaf of the foliation.

Invariant foliations appear naturally in the study of dynamical systems as the space of orbits of a flow, and as stable, unstable or centre manifolds in partially hyperbolic dynamical systems.

In this talk I will revisit some of the beautiful measurements of complexity for codimension one foliations, including their geometric entropy, transversal dynamics and irregular behavior for length averages.