The aim is to present a deeper overview of the long tree (not a linear sequence!) of works on the [local] geometry of flag-like and multi-flag-like control systems. From classical works of Engel, von Weber and E. Cartan, via [a non-exhaustive list:] Semple, Kumpera et al, Kennedy et al, Laumond-Risler-Jean, Montgomery-Zhitimirskii, Lejeune-Jalabert, till the actual investigations still in progress. And, time permitting, to bring in a couple of open questions.
Why flags? A kind of an answer is to be found in the benchmark von Weber's 1896 work. He started with a wish to go far beyond Frobenius to - possibly - all distributions. And eventually had to settle for the condition now known as `Goursat condition' (i.e., to settle for the 1-flags).
Special multi-flags, or - for some researchers - `Goursat multi-flags', started to be attacked by Kumpera-Rubin around the year 2000. The current state of the art is, in Kennedy et al 2017' words, as follows: "we seem to be very far from a full understanding of where and why moduli occur".