Diffractive processes are important non-perturbative phenomena of strong interaction, which are studied in various measurements at the LHC. Recent results of the CMS and TOTEM experiments are presented in this paper. First, the measurement of central exclusive and semiexclusive $\pi^+\pi^-$ production at 5.02 and 13 TeV is discussed. The total and differential cross sections of final states with $p_{\textsf{T}}(\pi) > 0.2$ GeV, $|\eta(\pi)| < 2.4$ are measured. In the second part of the paper, the measurement of the total and differential cross sections as functions of four-momentum transfer squared $t$ and proton fractional momentum loss $\xi$, in the $0.03 < |t| < 1.0$ GeV$^2$ and $0 < \xi < 0.1$ kinematic region, with at least two jets with $p_{\textsf{T}} > 40$ GeV and $|\eta| < 4.4$ is presented. This latter measurement utilized the proton tagging capabilities of the Roman Pot detectors of the TOTEM experiment.