The dynamic equation and coupling coefficient of the three-wave interaction among kinetic Alfvén waves (KAWs) are derived by use of plasma kinetic theory. Linear and nonlinear effects of finite ion Larmor radius are kept for arbitrary value of the ‘kinetic variable’ κ=k⊥ρi. The parametric decay KAW→KAW+KAWis investigated and the threshold amplitude for decay instability in a Maxwellian plasma is calculated. The growth rate of decay instability varies as k2⊥ in both limits κ2[Lt ]1 and κ2[Gt ]1. The main tendency of KAWs is towards nonlinear destabilization at very low wave amplitudes Bk/B0[lsim ]10−3. Two applications concerning KAW dynamics in the magnetosphere and in the solar corona show that three-wave resonant interaction among KAWs may be responsible for the turbulent character of their behaviour, often observed in space plasmas.