The first observation is reported of the combined production of three massive gauge bosons (VVV with V=W, Z) in proton-proton collisions at a center-of-mass energy of 13 TeV. The analysis is based on a data sample recorded by the CMS experiment at the CERN LHC corresponding to an integrated luminosity of 137 fb^{-1}. The searches for individual WWW, WWZ, WZZ, and ZZZ production are performed in final states with three, four, five, and six leptons (electrons or muons), or with two same-sign leptons plus one or two jets. The observed (expected) significance of the combined VVV production signal is 5.7 (5.9) standard deviations and the corresponding measured cross section relative to the standard model prediction is 1.02_{-0.23}^{+0.26}. The significances of the individual WWW and WWZ production are 3.3 and 3.4 standard deviations, respectively. Measured production cross sections for the individual triboson processes are also reported.