[en] In this paper, we describe the conceptual design for the suspension system for the test masses for Advanced LIGO, the planned upgrade to LIGO, the US laser interferometric gravitational-wave observatory. The design is based on the triple pendulum design developed for GEO 600 - the German/UK interferometric gravitational wave detector. The GEO design incorporates fused silica fibres of circular cross-section attached to the fused silica mirror (test mass) in the lowest pendulum stage, in order to minimize the thermal noise from the pendulum modes. The damping of the low-frequency modes of the triple pendulum is achieved by using co-located sensors and actuators at the highest mass of the triple pendulum. Another feature of the design is that global control forces acting on the mirrors, used to maintain the output of the interferometer on a dark fringe, are applied via a triple reaction pendulum, so that these forces can be implemented via a seismically isolated platform. These techniques have been extended to meet the more stringent noise levels planned for in Advanced LIGO. In particular, the Advanced LIGO baseline design requires a quadruple pendulum with a final stage consisting of a 40 kg sapphire mirror, suspended on fused silica ribbons or fibres. The design is chosen to aim to reach a target noise contribution from the suspension corresponding to a displacement sensitivity of 10^-19 m Hz^-1/2 at 10 Hz at each of the test masses