In this paper we extend the lazy narrowing calculus lnc of Middeldorp, Okui, and Ida [26] to conditional rewrite systems. The resulting lazy conditional narrowing calculus lcnc is highly non-deterministic. We investigate for which classes of conditional rewrite systems the completeness of lcnc is ensured. In order to improve the efficiency of the calculus, we pay special attention to the removal of non-determinism due to the selection of equations in goals by fixing a selection strategy. 1