The results of a new spectroscopic survey for high-redshift Lyman-limit absorption systems are presented. These results are based on recent observations of high-redshift QSOs by Lanzetta and coworkers, from which a total of 35 Lyman-limit systems from the spectra of 52 QSOs are identified. These new data are combined with existing observations of low- and high-redshift Lyman-limit systems in order to determine the properties of the absorbers over the redshift range 0.36 less-than-or-equal-to z less-than-or-equal-to 4.11. The most striking result to emerge from this new study is that on the basis of this data set the rate of incidence of the Lyman-limit systems with z greater-than-or-similar-to 2.5 is found to evolve strongly with redshift in the sense that the product of the number density per unit comoving volume and the absorption cross section increased with increasing redshift. Over the redshift range 2.5 less-than-or-similar-to z less-than-or-similar-to 3.7 the observed redshift distribution is well fitted by n(z) = n0(1 + z)gamma with gamma = 5.7 +/- 1.9. (In contrast, the rate of incidence of the Lyman-limit systems is virtually constant over the redshift range 0.35 less-than-or-equal-to z less-than-or-similar-to 2.5). If the observed evolution indicates intrinsic evolution of the absorbers, this result suggests that the evolution detected previously for the C IV-selected absorbers over a similar redshift range is most naturally interpreted as evolution of the ionization level of the absorbers rather than as a chemical enrichment effect, and model photoionization calculations indicate that the observed evolution of the absorption systems may be explained if the ionization parameters of the absorbers increases with decreasing redshift by a factor of almost-equal-to 3 within the redshift interval 2.5 less-than-or-similar-to z less-than-or-similar-to 3.7. This might arise as the result of a systematic evolution of the intensity of the diffuse background ultraviolet radiation field over the same redshift interval or might result from an increased contribution to the radiation field from "local" sources, such as would occur if the absorbers underwent a phase of massive star formation. The data are also used to investigate the H I column density distribution over the column density range 17.2 less-than-or-equal-to log [N(H I)/cm-2] less-than-or-equal-to 21.8 and to examine the multiple-component structure of the absorbing complexes. As in previous studies, the H I column density distribution is found to be well fitted by a power-law form, and an attempt is made to use model photoionization calculations to estimate the total column density distribution. The fractional displacement delta of the observed half-intensity point from the wavelength of the Lyman limit predicted from the absorption redshift is found to be correlated with the equivalent width of the C II lambda-1334 absorption line, which suggests that delta is available in a statistical sense as a probe of the multiple-component structure of the absorbing complexes. Implications of these results are discussed.