Recent STM experiments in BSCCO reveal modulations of the superconducting gap by 100%-200% on a lengthscale of a few lattice spacings. I will review the main properties of these modulations and argue that these results strongly suggest that the effective electron-electron interaction constant is  modulated in space by impurities, in case of BSCCO these impurities are the interstitial dopant oxygen atoms.

When magnetic field is applied, a new source of inhomogeneities appears, as the vortex lattice develops. In the second part of the talk I will describe the experimental data and the current state of the theory for the quasiparticles moving in the presence of vortices. The low energy part of the spectrum exhibits anomalous non-perturbative oscillations due to a singular nature of the quasiparticle wavefunctions near vortices, affecting a number of measurable quantities such as the quantized thermal Hall conductivity.