  We study Fano 3-folds with Fano index 2: that is, 3-folds X with rank Pic(X)
= 1, Q-factorial terminal singularities and -K_X = 2A for an ample Weil divisor
A. We give a first classification of all possible Hilbert series of such
polarised varieties X,A and deduce both the nonvanishing of H^0(X,-K_X) and the
sharp bound (-K_X)^3 >= 8/165. We list families that can be realised in
codimension up to 4.
