The electronic structure of a quasiperiodic superlattice with slabs of A- and B-type layers is studied within a tight-binding model by use of the transfer matrix and a perturbative approach. It is found that the band gaps vary strongly within the two-dimensional Brillouin zone and, in the case of more than one layer in a slab, there are a few eigenvalues for each k? for which the eigenstates appear to be extended in the quasiperiodic direction. The latter is true even for the Fibonacci chain with more than one atom per cell. {\textcopyright} 1987 The American Physical Society.