A comprehensive phase-shift analysis of π+-p elastic-scattering data at 310-Mev incident-pion laboratory kinetic energy has been performed. The experimental data utilized include measurements of the differential and total cross sections and of the recoil-proton polarization. The D-wave phase shifts were found to be definitely needed in order to attain an adequate fit to the data. A general search for phase-shift solutions was carried out, using S-, P-, and D-wave phase shifts. One solution-of the Fermi type-was found that fits the data significantly better than any of the other solutions obtained. The calculated errors in the phase shifts of this set vary from 0.4 to 0.6 deg. Because it was felt that these errors might be deceivingly restrictive, the effects of small nuclear F-wave phase shifts on the results of the analysis were investigated and were found to be large: not only are the uncertainties in the original Fermi-type solution increased, but additional sets of phase shifts arise that fit the data well. One of these new solutions is similar to the original Fermi set except that the magnitudes of the phase shifts in this new fit are in general larger than those in the initial solution, and the signs of the D-wave phase shifts are reversed. The nuclear phase shifts in the original Fermi solution and their rms errors are (when F-wave phase shifts are allowed): S3,1=-17.2+/-2.6 deg, P3,1=-2.9+/-4.0 deg, P3,3=135.0+/-0.6 deg, D3,3=3.1+/-2.6 deg, D3,5=-4.9+/-2.1 deg, F3,5=0.5+/-0.6 deg, F3,7=-0.6+/-1.4 deg. Although theory appears to favor this set, further theoretical and experimental evidence is desirable. The values given here for the first five phase shifts approximate the corresponding values obtained when the F-wave phase shifts were assumed negligible. However, all except P3,3 fall outside the limits set by the small original errors. Inelastic-scattering processes were neglected during the phase-shift analysis. Calculations indicate that, if these processes could properly be taken into account, any changes in the quoted values of the phase shifts would probably be well within the corresponding errors given here. Extension of the phase-shift inquiries to include G waves was attempted, but it was observed that the available data and theory do not allow the G-wave interaction to be significantly incorporated into the analysis.