We study generic quadric sections of irreducible space curves. Suitable examples help to understanding the substantial differences of behaviour with respect to generic plane sections of irreducible curves. For instance we produce an example of points on a smooth quadric in strong uniform position whose Hilbert function cannot be as the one of a generic quadric section of any irreducible curve. In order to try to find all possible Hilbert functions of generic quadric sections of irreducible curves we characterize sequences of positive integers which are the first difference of the Hilbert funtions of generic quadric sections of ACM curves and then of maximal rank Buchsbaum curves.