In this paper we consider the finite-volume-element method for general second-order quasilinear elliptic problems over a convex polygonal domain in the plane. Using reasonable assumptions, we show the existence and uniqueness of the finite-volume-element approximations. It is proved that the finite-volume-element approximations are convergent with , where r > 2, and in the H1-, W1, ∞- and L2-norms, respectively, for u ∈ W2, r(Ω) and u ∈ W2, ∞(Ω) ∩ W3, p(Ω), where p > 1. Moreover, the optimal-order error estimates in the W1, ∞- and L2-norms and an estimate in the L∞-norm are derived under the assumption that u ∈ W2, ∞(Ω) ∩ H3(Ω). Numerical experiments are presented to confirm the estimates.