The Volterra convolution operator [Vscr ]f(x) = ∫x0ϕ (x−y)f(y)dy, where ϕ(·) is a non-negative non-decreasing integrable kernel on [0, 1], is considered. Under certain conditions on the kernel ϕ, the maximal Banach function space on [0, 1] on which the Volterra operator is a continuous linear operator with values in a given rearrangement invariant function space on [0, 1] is identified in terms of interpolation spaces. The compactness of the operator on this space is studied.