  In this Chapter, we present some interesting properties of quantum walks on
the line. We concentrate our attention in the emergence of invariance and
provide some insights into the ultimate origin of the observed behavior. In the
first part of the Chapter, we review the building blocks of the
quantum-mechanical version of the standard random walk in one dimension. The
most distinctive difference between random and quantum walks is the replacement
of the random coin in the former by the action of a unitary operator upon some
internal property of the later. We provide explicit expressions for the
solution to the problem when the most general form for the homogeneous unitary
operator is considered, and we analyze several key features of the system as
the presence of symmetries or stationary limits. After that, we analyze the
consequences of letting the properties of the coin operator change from site to
site, and from time step to time step. In spite of this lack of homogeneity,
the probabilistic properties of the motion of the walker can remain unaltered
if the coin variability is chosen adequately. Finally, we show how this
invariance can be connected to the gauge freedom of electromagnetism.
