Extrapolation methods that accelerate the convergence of coupled-cluster energy sequences toward the full configuration-interaction (FCI) limit are developed and demonstrated for a variety of atoms and small molecules for which FCI energies are available, and the results are compared with those from Moller-Plesset (MP) perturbation theory. For the coupled-cluster sequence SCF, CCSD, CCSD(T), a method based on a continued-fraction formalism is found to be particularly successful. It yields sufficient improvement over conventional CCSD(T) that the results become competitive with, and often better than, results from the MP4-qlambda method (MP4 summed with quadratic approximants and lambda transformation). The sequence SCF, CCSD, CCSDT can be extrapolated with a quadratic approximant but the results are not appreciably more accurate than those from the CCSD(T) continued fraction. Singularity analysis of the MP perturbation series provides a criterion for estimating the accuracy the CCSD(T) continued fraction.