We show how XAFS has been successfully used in the determination of th
e local crystal structure of high-temperature superconductors, with ad
vantages over traditional diffraction techniques. We review the experi
mental results that yielded the first evidence for an axial-oxygen-cen
tered lattice instability connected with the superconductivity transit
ion. The interpretation of this instability in terms of a dynamical tu
nneling model suggests the presence of polarons in these materials. XA
FS on Tl2Ba2CuO6 and other Tl-based superconductors indicate the prese
nce of local instabilities in the CuO2 planes of these materials, in a
ddition to axial-oxygen instabilities.