This review is concerned with the experimental data on elastic scattering of mesons and nucleons by protons. Theoretical ideas about hadronic interactions at high energies are discussed. An exposition is given of the techniques for measuring differential cross sections in the region of very small momentum transfers. A summary is given of the information which has been obtained in various experiments on the total cross sections for $\pi^\pm N$, $K^\pm N$, $NN$, and $\bar N N$ interactions, the real parts of the elastic scattering amplitudes at $t=0$, the slopes of the diffraction peaks, large-angle scattering, and polarization effects in elastic scattering reactions. It is observed that the entire set of experimental data is consistent with dispersion relations and asymptotic theorems. The slope of the diffraction peak exhibits a systematic (approximately logarithmic) growth at high energies. This growth corresponds to a relatively small value for the slope of the Pomeranchuk trajectory, $\alpha_{\mathrm{P}'}(0)\approx$0.3 GeV$^{-2}$. The experimental data in elastic scattering processes and charge-exchange reactions are in agreement with the predictions of complex angular momentum theory. A discussion is given of new phenomena which have been observed using the accelerators at Serpukhov and Batavia and the CERN intersecting storage rings: a growth of the total cross sections for hadronic interactions, a change of sign of the real part of the zero-angle elastic scattering amplitude, and other effects which show up at energies $\gtrsim$100 GeV.