  A measurement of the W$^+$W$^-$ boson pair production cross section in
proton-proton collisions at $\sqrt{s} =$ 13 TeV is presented. The data used in
this study are collected with the CMS detector at the CERN LHC and correspond
to an integrated luminosity of 35.9 fb$^{-1}$. The W$^+$W$^-$ candidate events
are selected by requiring two oppositely charged leptons (electrons or muons).
Two methods for reducing background contributions are employed. In the first
one, a sequence of requirements on kinematic quantities is applied allowing a
measurement of the total production cross section: 117.6 $\pm$ 6.8 pb, which
agrees well with the theoretical prediction. Fiducial cross sections are also
reported for events with zero or one jet, and the change in the zero-jet
fiducial cross section with the jet transverse momentum threshold is measured.
Normalized differential cross sections are reported within the fiducial region.
A second method for suppressing background contributions employs two random
forest classifiers. The analysis based on this method includes a measurement of
the total production cross section and also a measurement of the normalized jet
multiplicity distribution in W$^+$W$^-$ events. Finally, a dilepton invariant
mass distribution is used to probe for physics beyond the standard model in the
context of an effective field theory, and constraints on the presence of
dimension-6 operators are derived.
