  Albertson has defined the irregularity of a simple undirected graph $G=(V,E)$
as $ \irr(G) = \sum_{uv\in E}|d_G(u)-d_G(v)|,$ where $d_G(u)$ denotes the
degree of a vertex $u \in V$. Recently, this graph invariant gained interest in
the chemical graph theory, where it occured in some bounds on the first and the
second Zagreb index, and was named the third Zagreb index Fath-Tabar. For
general graphs with $n$ vertices, Albertson has obtained an asymptotically
tight upper bound on the irregularity of $4 n^3 /27.$ Here, by exploiting a
different approach than in Albertson, we show that for general graphs with $n$
vertices the upper bound $\lfloor \frac{n}{3} \rfloor \lceil \frac{2 n}{3}
\rceil (\lceil \frac{2 n}{3} \rceil -1)$ is sharp. Next, we determine
$k$-cyclic graphs with maximal irregularity. We also present some bounds on the
maximal/minimal irregularity of graphs with fixed minimal and/or maximal vertex
degrees, and consider an approximate computation of the irregularity of a
graph.
