A linear space is a point-line incidence structure such that each pair of points is incident to exactly one line. It is regular if all lines are incident to the same number of points. (This is equivalent to a balanced block design with $\lambda=1$.)
I will discuss linear spaces which admit a very special class of groups of automorphisms, called extremely primitive groups. Together with Melissa Lee, we have almost finished the classification of these linear spaces. In the process, we discovered some new interesting constructions which I will describe.