An introduction to Engel topology

Maximally non-integrable plane fields in 4-dimensional manifolds are called Engel structures. We think of them as analogues of contact structures in dimension 4 and, indeed, in the last decase we have seen many developments that indicate that there are strong parallels between the two theories.
My goal in this talk will be to walk you through these recent results. Concretely, I will explain what we know about the classification up to homotopy of Engel structures and their submanifolds. I will assume no familiarity with these concepts, so I will begin with the basic examples. I will finish the talk with an overview of the main open questions in the field.