Kelvin-Helmholtz instability (KHI) is a basic physical process in fluids and magnetized plasmas, with applications successfully modelling e.g. exponentially growing instabilities observed at magnetospheric and heliospheric boundaries, in the solar or Earth's atmosphere and within astrophysical jets. Here, we report the discovery of the KHI in solar blowout jets and analyse the detailed evolution by employing high-resolution data from the Interface Region Imaging Spectrograph (IRIS) satellite launched in 2013. The particular jet we focus on is rooted in the surrounding penumbra of the main negative polarity sunspot of Active Region 12365, where the main body of the jet is a super-penumbral structure. At its maximum, the jet has a length of 90 Mm, a width of 19.7 Mm, and its density is about 40 times higher than its surroundings. During the evolution of the jet, a cavity appears near the base of the jet, and bi-directional flows originated from the top and bottom of the cavity start to develop, indicating that magnetic reconnection takes place around the cavity. Two upward flows pass along the left boundary of the jet successively. Next, KHI develops due to a strong velocity shear (∼204 km s-1) between these two flows, and subsequently the smooth left boundary exhibits a sawtooth pattern, evidencing the onset of the instability.