The normal covering number γ(G) of a finite, non-cyclic group G is the least number of proper subgroups such that each element of G lies in some conjugate of one of these subgroups. We prove that there is a positive constant c such that, for G a symmetric group Sym(. n) or an alternating group Alt(. n), γ(. G) ≥ c n. This improves results of the first two authors who had earlier proved that aφ(. n) ≤ γ(. G) ≤ 2. n/3, for some positive constant a, where φ is the Euler totient function. Bounds are also obtained for the maximum size κ(. G) of a set X of conjugacy classes of G = Sym(. n) or Alt(. n) such that any pair of elements from distinct classes in X generates G, namely c n ≤ κ(. G) ≤ 2. n/3. © 2013 Elsevier Inc.