  The power graph $\mathcal{P}(G)$ of a group $G$ is the graph whose vertex set
is $G$, having an edge between two distinct vertices if one is the power of the
other. The directed power graph $\vec{\mathcal{P}}(G)$ of a group $G$ is the
digraph whose vertex set is $G$, having an arc from $x$ to $y$, with $x\ne y$,
whenever $y$ is a power of $x$. We rewrite two Cameron's articles concerning
the reconstruction of $\vec{\mathcal{P}}(G)$ from $\mathcal{P}(G)$. We correct
mistakes that appear in the papers. In particular, we add missing cases needed
to complete the main theorems of these articles.
  We also study the quotient of the power graph under some equivalence
relations. We close the thesis with lower bounds for the maximum length of a
cycle in the power graph of a group.
