Combinatorial t-designs are useful combinatorial objects, borne out of statistical experiment needs. They consist in points and subsets of points called blocks, such that each block has the same size and every subset of t points is in the same number of blocks. I will explain several transitivity properties that the automorphism group of a t-design can have. Then I will concentrate on 2-designs admitting a point-imprimitive group of automorphisms which is either flag-transitive or block-transitive. I will survey some known results, and explain some recent results from 3 papers on this topic. Joint work with Cheryl Praeger, Carmen Amarra, Hongxue Liang, and Binzhou Xia.