Extending results of Bauer, Catanese and Grunewald, and of Fuertes and
Gonz\'alez-Diez, we show that Beauville surfaces of unmixed type can be
obtained from the groups L_2(q) and SL_2(q) for all prime powers q>5, and the
Suzuki groups Sz(2^e) and the Ree groups R(3^e) for all odd e>1. We also show
that L_2(q) and SL_2(q) admit strongly real Beauville structures, yielding real
Beauville surfaces, if and only if q=8 or q>9.