The electromagnetic form factors of charged and neutral kaons are strongly constrained by their lowenergy singularities, in the isovector part from twopion intermediate states and in the isoscalar contribution in terms of ω and φ residues. The former can be predicted using the respective ππ → KK partialwave amplitude and the pion electromagnetic form factor, while the latter parameters need to be determined from electromagnetic reactions involving kaons. We present a global analysis of time-and spacelike data that implements all of these constraints. The results enable manifold applications: kaon charge radii, elastic contributions to the kaon electromagnetic self energies and corrections to Dashen's theorem, kaon boxes in hadronic light-by-light (HLbL) scattering, and the φ region in hadronic vacuum polarization (HVP). Our main results are: r 2 c = 0.359(3) fm 2 , r 2 n = −0.060(4) fm 2 for the charged and neutral radii, = 0.63( 40) for the elastic contribution to the violation of Dashen's theorem, a K-box µ = −0.48(1) × 10 −11 for the charged kaon box in HLbL scattering, and a HVP µ [K + K − , ≤ 1.05 GeV] = 184.5(2.0) × 10 −11 , a HVP µ [K S K L , ≤ 1.05 GeV] = 118.3(1.5) × 10 −11 for the HVP integrals around the φ resonance. The global fit to KK gives Mφ = 1019.479(5) MeV, Γφ = 4.207(8) MeV for the φ resonance parameters including vacuumpolarization effects.