  The first observation of electroweak production of same-sign W boson pairs in
proton-proton collisions is reported. The data sample corresponds to an
integrated luminosity of 35.9 fb$^{-1}$ collected at a center-of-mass energy of
13 TeV with the CMS detector at the LHC. Events are selected by requiring
exactly two leptons (electrons or muons) of the same charge, moderate missing
transverse momentum, and two jets with a large rapidity separation and a large
dijet mass. The observed significance of the signal is 5.5 standard deviations,
where a significance of 5.7 standard deviations is expected based on the
standard model. The ratio of measured event yields to that expected from the
standard model at leading-order is 0.90 $\pm$ 0.22. A cross section measurement
in a fiducial region is reported. Bounds are given on the structure of quartic
vector boson interactions in the framework of dimension-eight effective field
theory operators and on the production of doubly charged Higgs bosons.
