A relational structure A with a countable universe is defined to be homogeneous iff every finite partial isomorphism of A can be extended to an automorphism of A. Endow the universe of A with the discrete topology. Then the automorphism group Aut(A) of A becomes a topological group (with the subspace topology inherited from the suitable topological power of the discrete topology on A). Recall, that a tuple 0 , ..., n−1 of elements of Aut(A) is defined to be weakly generic iff its diagonal conjugacy class (in the group theoretic sense) is dense in the topological sense, and further, the 0 , ..., n−1-orbit of each a ∈ A is finite. Investigations about weakly generic automorphisms have model theoretic origins (and reasons); however, the existence of weakly generic automorphisms is closely related to interesting results in finite combinatorics, as well. In this work we survey some connections between the existence of weakly generic automorphisms and finite combinatorics, group theory and topology. We will recall some classical results as well as some more recently obtained ones.