Anderson localization-delocalization (LD) transitions separate insulating (localized) and metallic (delocalized) states of non-interacting quantum particles moving in random potentials. Wave functions at Anderson transitions are known to exhibit complicated scale-invariant behavior best characterized by an infinite set of multifractal exponents. After introducing and explaining these concepts, I will present our recent study of multifractal spectra of critical wave functions at various Anderson transitions, focusing on finite systems with boundaries. In two dimensions the boundary behavior of the critical wave functions provides a way to study the problem of conformal invariance at these transitions. One of the more mysterious LD transitions is the so called integer quantum Hall transition, for which the multifractal spectra were conjectured to be exactly parabolic in a number of theoretical proposals. Our numerical results for the Chalker-Coddington network of this transition model firmly rule out the exact parabolicity.