  Feshbach resonances in ultracold collisions often result from an interplay
between many collision channels. Simple two-channel models can be introduced to
capture the basic features, but cannot fully reproduce the situation when
several resonances from different closed channels contribute to the scattering
process. Using the formalism of multichannel quantum defect theory we develop
an analytical model of overlapping Feshbach resonances. We find a general
formula for the variation of the scattering length with magnetic field in the
vicinity of an arbitrary number of resonances, characterized by simple
parameters. Our formula is in excellent agreement with numerical coupled
channels calculations for several cases of overlapping resonances in the
collisions of two $^7$Li atoms or two Cs atoms.
