  The abundance of large and heterogeneous systems is rendering contemporary
data more pervasive, intricate, and with a non-regular structure. With
classical techniques facing troubles to deal with the irregular (non-Euclidean)
domain where the signals are defined, a popular approach at the heart of graph
signal processing (GSP) is to: (i) represent the underlying support via a graph
and (ii) exploit the topology of this graph to process the signals at hand. In
addition to the irregular structure of the signals, another critical limitation
is that the observed data is prone to the presence of perturbations, which, in
the context of GSP, may affect not only the observed signals but also the
topology of the supporting graph. Ignoring the presence of perturbations, along
with the couplings between the errors in the signal and the errors in their
support, can drastically hinder estimation performance. While many GSP works
have looked at the presence of perturbations in the signals, much fewer have
looked at the presence of perturbations in the graph, and almost none at their
joint effect. While this is not surprising (GSP is a relatively new field), we
expect this to change in the upcoming years. Motivated by the previous
discussion, the goal of this thesis is to advance toward a robust GSP paradigm
where the algorithms are carefully designed to incorporate the influence of
perturbations in the graph signals, the graph support, and both. To do so, we
consider different types of perturbations, evaluate their disruptive impact on
fundamental GSP tasks, and design robust algorithms to address them.
