In this talk we give a survey on a certain number of multi-parameter structures, on $\mathbb{R}^n$ and on nilpotent groups, that have first appeared in joint work of mine with A. Nagel, E. Stein and S. Wainger. They include flag and multi-norm structures.

These structures are intermediate between the one-parameter dilation structures of standard Calderón-Zygmund theory and the full n-parameter product structure. Each structure has its own type of maximal functions, singular integral operators, square functions, Hardy spaces.