The NA62 experiment at CERN collected the world's largest data set of charged kaon decays in 2016-2018, leading to the most precise measurement of the branching fraction of the ultra-rare K+→π+νν‾ decay, based on 20 candidates. This provides evidence for the observation of the K+→π+νν‾ decay with a significance of 3.4σ. This measurement is also used to set limits on B(K+→π+X), where X is a scalar or pseudo-scalar particle. The analysis of the full 2016-2018 data sample is reviewed. More recent results from NA62 analyses of K+→π0e+νγ, K+→π+μ+μ− and K+→π+γγ decays, using data samples recorded in 2017–2018, are also reported. The radiative kaon decay K+→π0e+νγ (Ke3γ) is studied with a data sample of O(100k) Ke3γ candidates with sub-percent background contamination. Final results of the most precise measurements of the Ke3γ branching ratios and T-asymmetry are presented. The K+→π+μ+μ− sample comprises about 27k signal events with negligible background contamination, and the results presented include the most precise determination of the branching fraction and the decay form factor. The K+→π+γγ sample contains about 4k signal events with 10% background contamination, and the preliminary analysis results improve the precision of the branching fraction measurement by a factor of 3 with respect to previous measurements. An overview of the latest NA62 results and future prospects for the experiment is presented.