  We study the critical depinning current J_c, as a function of the applied
magnetic flux Phi, for quasiperiodic (QP) pinning arrays, including
one-dimensional (1D) chains and two-dimensional (2D) arrays of pinning centers
placed on the nodes of a five-fold Penrose lattice. In 1D QP chains of pinning
sites, the peaks in J_c(Phi) are shown to be determined by a sequence of
harmonics of long and short periods of the chain. This sequence includes as a
subset the sequence of successive Fibonacci numbers. We also analyze the
evolution of J_c(Phi) while a continuous transition occurs from a periodic
lattice of pinning centers to a QP one; the continuous transition is achieved
by varying the ratio gamma = a_S/a_L of lengths of the short a_S and the long
a_L segments, starting from gamma = 1 for a periodic sequence. We find that the
peaks related to the Fibonacci sequence are most pronounced when gamma is equal
to the "golden mean". The critical current J_c(Phi) in QP lattice has a
remarkable self-similarity. This effect is demonstrated both in real space and
in reciprocal k-space. In 2D QP pinning arrays (e.g., Penrose lattices), the
pinning of vortices is related to matching conditions between the vortex
lattice and the QP lattice of pinning centers. Although more subtle to analyze
than in 1D pinning chains, the structure in J_c(Phi) is determined by the
presence of two different kinds of elements forming the 2D QP lattice. Indeed,
we predict analytically and numerically the main features of J_c(Phi) for
Penrose lattices. Comparing the J_c's for QP (Penrose), periodic (triangular)
and random arrays of pinning sites, we have found that the QP lattice provides
an unusually broad critical current J_c(Phi), that could be useful for
practical applications demanding high J_c's over a wide range of fields.
