  The aim of this thesis is to investigate how the structure of a digraph
affects its dichromatic number and to extend various results on undirected
colouring to digraphs.
  In the first part of this thesis, we examine how the dichromatic number
interacts with other metrics. First, we consider the degree, which is the
maximum number of neighbours of a vertex. In the undirected case, this
corresponds to Brooks' theorem, a celebrated theorem with multiple variations
and generalizations. In the directed case, there is no natural metric
corresponding to the maximum degree, so we explore how different notions of
maximum directed degree lead to either Brooks-like theorems or impossibility
results. We also investigate the maximum local-arc connectivity, a metric that
encompasses several degree-like metrics.
  The second part of this manuscript focuses on a directed analogue of the
Gy\'arf\'as-Sumner conjecture. The Gy\'arf\'as-Sumner conjecture tries to
characterize sets S of undirected graphs such that graphs with large enough
chromatic number must contain a graph of S. This conjecture is still largely
open. On digraphs, a corresponding conjecture was proposed by Aboulker,
Charbit, and Naserasr. We prove several subcases of this conjecture, mainly
demonstrating that certain classes of digraphs have bounded dichromatic number.
  In the last part of this thesis, we address the d-edge-defective-colouring
problem, which involves colouring edges of a multigraph such that, for any
vertex, no colour appears on more than d of its incident edges. When d equals
one, this corresponds to the infamous edge-colouring problem. Shannon
established a tight bound on the number of colours needed relative to the
maximum degree when d equals one, and we extend this result to any value of d.
We also explore this problem on simple graphs and prove results that extend
Vizing's theorem to any value of d.
