  The quantum walk (QW) is the term given to a family of algorithms governing
the evolution of a discrete quantum system and as such has a founding role in
the study of quantum computation. We contribute to the investigation of QW
phenomena by performing a detailed numerical study of discrete-time quantum
walks. In one dimension (1D), we compute the structure of the probability
distribution, which is not a smooth curve but shows oscillatory features on all
length scales. By analyzing walks up to $N$ = 1000000 steps, we discuss the
scaling characteristics and limiting forms of the QW in both real and Fourier
space. In 2D, with a view to ready experimental realization, we consider two
types of QW, one based on a four-faced coin and the other on sequential
flipping of a single two-faced coin. Both QWs may be generated using two
two-faced coins, which in the first case are completely unentangled and in the
second are maximally entangled. We draw on our 1D results to characterize the
properties of both walks, demonstrating maximal speed-up and emerging
semi-classical behavior in the maximally entangled QW.
