  The NA62 experiment reports the branching ratio measurement BR$(K^+
\rightarrow \pi^+ \nu\bar{\nu}) = (10.6^{+4.0}_{-3.4} |_{\rm stat} \pm 0.9_{\rm
syst}) \times 10 ^{-11}$ at 68% CL, based on the observation of 20 signal
candidates with an expected background of 7.0 events from the total data sample
collected at the CERN SPS during 2016-2018. This provides evidence for the very
rare $K^+ \rightarrow \pi^+ \nu\bar{\nu}$ decay, observed with a significance
of 3.4$\sigma$. The experiment achieves a single event sensitivity of
$(0.839\pm 0.054)\times 10^{-11}$, corresponding to 10.0 events assuming the
Standard Model branching ratio of $(8.4\pm1.0)\times10^{-11}$. This measurement
is also used to set limits on BR($K^+ \to \pi^+ X$), where $X$ is a scalar or
pseudo-scalar particle. Details are given of the analysis of the 2018 data
sample, which corresponds to about 80% of the total data sample.
