The earliest searches for the $K^+ \rightarrow\pi^+ \nu \bar{\nu}$ decay date back to 1969, when a bubble chamber experiment at the Argonne National Laboratory of Michigan defined a first upper limit to its branching ratio \cite{kpinunu69}:
\begin{align}
&BR(K^+ \rightarrow\pi^+ \nu \bar{\nu}) < 10^{-4} \quad\qquad \text{(1969)}
\end{align}

Four years later, the limit was improved to $BR(K^+ \rightarrow\pi^+ \nu \bar{\nu}) < 5.6 \cdot 10^{-7}$ by a spark chamber experiment at the Berkeley Bevatron \cite{kpinunu73}, followed by a search at the KEK Proton Synchrotron that yielded \cite{kpinunuKek}
\begin{align}
&BR(K^+ \rightarrow\pi^+ \nu \bar{\nu}) < 1.4 \cdot 10^{-7} \qquad \text{(1981)}
\end{align}

Since the 80's, a large effort has been devoted at the Brookhaven National Laboratory to the study of rare, ultra-rare and forbidden kaon decays. The E787 collaboration published a first measurement based on 3 events interpreted as $K^+ \rightarrow\pi^+ \nu \bar{\nu}$ decays \cite{kpinunu787}:
\begin{align}
&BR(K^+ \rightarrow\pi^+ \nu \bar{\nu}) = 1.47^{+1.30}_{-0.89} \cdot 10^{-10} \qquad \text{(2004)}
\end{align}
%where the first error accounts for the experimental systematics and the second represents the pure statistic uncertainty.\\

The follow-up experiment E949 was able to collect 7 more candidate events with an estimated background of $ 0.93^{+0.32}_{-0.24}\pm0.17 $ events, where the first error accounts for the experimental systematics and the second represents the pure statistic uncertainty, leading to a combined result of \cite{kpinunu949,Artamonov:2008qb}
\begin{align}
&BR(K^+ \rightarrow\pi^+ \nu \bar{\nu}) = 1.73^{+1.15}_{-1.05} \cdot 10^{-10} \qquad \text{(2009)}
\end{align}
which is consistent with the Standard Model expectations, within the large statistical errors.\\

It is important to note that every $K \rightarrow\pi \nu \bar{\nu}$ experiment performed up to now has used low-energy stopped-kaon beams. NA62 will instead employ a high-energy beam, thus studying in-flight kaon decays.