A new recursive algorithm for the solution of the problem of scattering of light (of an arbitrarily polarized plane electromagnetic wave) by multilayer confocal spheroidal particles is constructed. This approach preserves the advantages of the two approaches proposed earlier by us for single-layer and two-layer spheroids (special choice of scalar potentials and utilization of the basis of wave spheroidal harmonics) and for homogeneous axially symmetric particles (formulation of the problem in terms of surface integral equations, calculation of the potentials inside the particle from the potentials of the incident radiation, and calculation of the potentials of the scattered radiation from the potentials inside the particle). In the case of multilayer particles, the potential inside each shell is a sum of two terms. The first has the properties of the incident radiation (no singularities inside the volume enclosed by the external boundary of the shell), whereas the second term has the properties of the scattered radiation (satisfies the radiation conditions at infinity). Therefore, as the calculation progresses from one layer to the next (from the core to the outer shell), the dimensionality of the reduced linear matrix equations for the unknown expansion coefficients of the scattered field potentials does not increase with respect to the case of a homogeneous spheroid. The algorithm is particularly simple and lucid (as far as possible for such a complex problem). In the case of spherical multilayer particles, the solution can be found explicitly.