We describe the design of the ESO Adaptive Optics (AO) systems for the very large telescope interferometer (VLTI). We consider hereafter both the tip-tilt only corrections and the high order systems. The high order AO systems are designed for K-band operation on the Unit Telescopes (UT). The K-band UT beams will be combined with the 1.8m Auxiliary Telescopes (AT) operating with tip-tilt correction only, via ESO's Strap system. The UT-AO system will be hosted in the Coude' laboratory, with the deformable mirror inserted at the M8 location of the optical train. The wavefront sensor retains the option to be either in the Coude' lab, before the delay lines, or at the end of the beam combining path in the Interferometry laboratory, depending on the instrument attached and its use.