We have investigated 15 excited states of the allyl radical, including the lowest three valence states (two doublets and one quartet) and the n = 3 Ry series, using coupled cluster methods that approximate the correlation effects of connected triple excitations. The quality of the excitation energies is measured on the basis of comparison to existing theoretical and experimental data, as well as on the basis of three diagnostics related to spin contamination and the overall level of excitation of a given state. Basis-set effects are significant for states exhibiting substantial Rydberg character, and the use of molecule-centered diffuse functions appears to provide an accurate description of such states, while avoiding the computational expense of basis sets in which diffuse functions are added to every atom in the molecule. In contrast to earlier observations for linear carbon-chain radicals, coupled cluster methods compare well to both theoretical predictions and experimental band origins, where discrepancies in the latter are sometimes attributable to structural relaxation in the excited state. One of the three lowest (2)B(1) excited states exhibits a twisting of the terminal methylene groups to yield a C(2)-symmetry minimum. The most challenging states for coupled cluster methods are of A(2) symmetry, where both spin contamination and basis-set effects are appreciable.