The first observation is reported of the combined production of three massive gauge bosons (
V
V
V
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
W
W
W
,
W
W
Z
,
W
Z
Z
, and
Z
Z
Z
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
V
V
V
production signal is 5.7 (5.9) standard deviations and the corresponding measured cross section relative to the standard model prediction is
1.02
+
0.26
−
0.23
. The significances of the individual
W
W
W
and
W
W
Z
production are 3.3 and 3.4 standard deviations, respectively. Measured production cross sections for the individual triboson processes are also reported.