We pose the problem of identifying the set K (G, Ω) of Galois number fields with given Galois group G and root discriminant less than the Serre constant Ω ≈ 44.7632. We definitively treat the cases G = A4, A5, A6 and S4, S5, S6, finding exactly 59, 78, 5 and 527, 192, 13 fields, respectively. We present other fields with Galois group SL3 (2), A7, S7, PGL2 (7), SL2 (8), Σ L2 (8), PGL2 (9), P Γ L2 (9), PSL2 (11), and A52 . 2, and root discriminant less than Ω. We conjecture that for all but finitely many groups G, the set K (G, Ω) is empty.