  Results by the CMS Collaboration on forward physics, diffraction, and physics
in the small-$x$ limit of quantum chromodynamics (QCD), are presented. In
particular, results on azimuthal angle decorrelations between two jets in
events where two outermost jets are separated by a large rapidity interval are
discussed. In addition, results based on the production of two jets separated
by a large rapidity gap (interval void of radiation) are presented. These dijet
production processes are expected to be sensitive to
Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution effects. We highlight results on
inclusive forward jet production and on exclusive vector meson production in
proton-lead collisions, which access gluon densities in the small-$x$ and $Q^2$
where saturation effects may play a role. A summary of results on underlying
event activity studies based on inclusive $Z$ boson production, charged
particle spectra in minimum bias events, and energy density in forward
pseudorapidities is presented. These studies provide valuable inputs for Monte
Carlo event generator tuning, and test predictions based on perturbative and
non-perturbative QCD techniques.
