The confinement of a polymer into a small space is thermodynamically unfavorable because of the reduction in the number of conformational states. The entropic penalty affects a variety of biological processes, and it plays an important role in polymer transport properties and in microfluidic devices. We determine the entropic penalty for the confinement of elastic polymer of persistence length P in the long-chain limit. We examine three geometries: (1) parallel planes separated by a distance d (a slit); (2) a circular tube of diameter d; and (3) a sphere of diameter d. We first consider infinitely thin (ideal) chains. As d/P drops from 100 to 0.01, T Delta S rises from similar to 5 x 10(-4)kT to similar to 30kT per persistence length for cases (I) and (2), with the entropic penalty for case (2) being consistently about twice that for case (1). T Delta S is similar to 5kT per persistence length for confinement to a sphere when d = P, about twice the value predicted by mean field theory. For all three geometries, in the limit d/P >> 1, the asymptotic behavior of Delta S vs d is consistent with the d(-2) behavior predicted by theory. In the limit d/P << 1, the scaling of Delta S for slits and tubes is also consistent with earlier predictions (d(-2/3)). Finally, we treat volume exclusion effects, examining chains of diameter D > 0. Confinement to a narrow slit or tube (d/P << 1) has the same entropic penalty as that for an ideal chain in a slit or tube with d' = d - D; in the weak confinement regime (d/P >> 1), the entropic penalties are significantly larger than those for infinitely thin chains. When a chain of finite diameter is forced into a sphere or other closed cavity, the entropic confinement penalty rises without limit because there are no configurations available to the chain once its volume exceeds that of the cavity.