It is well known that for minimal surfaces of general type with q = p_g=1 the invariant K^2 can take values K^2= 2, .. 9.

The fine classification for K^2=2,3 was achieved some years ago, and it required the development of several tools, in joint works with Ciro Ciliberto and Roberto Pignatelli.

Existence for each value of K^2 was shown through the work of several authors , which I shall briefly survey. I shall then describe several recent results, obtained with Ingrid Bauer and Davide Frapporti, and then I shall speak on work in progress with Toledo, Stover and Keum on the geometric construction of the case K^2=9, whose existence was shown by Cartwright and Steger using computer calculations.