  The {\it inversion} of a set $X$ of vertices in a digraph $D$ consists in
reversing the direction of all arcs of $D\langle X\rangle$. The {\it inversion
number} of an oriented graph $D$, denoted by ${\rm inv}(D)$, is the minimum
number of inversions needed to transform $D$ into an acyclic oriented graph. In
this paper, we study a number of problems involving the inversion number of
oriented graphs. Firstly, we give bounds on ${\rm inv}(n)$, the maximum of the
inversion numbers of the oriented graphs of order $n$. We show $n -
\mathcal{O}(\sqrt{n\log n}) \ \leq \ {\rm inv}(n) \ \leq \ n - \lceil \log
(n+1) \rceil$. Secondly, we disprove a conjecture of Bang-Jensen et al.
asserting that, for every pair of oriented graphs $L$ and $R$, we have ${\rm
inv}(L\Rightarrow R) ={\rm inv}(L) + {\rm inv}(R)$, where $L\Rightarrow R$ is
the oriented graph obtained from the disjoint union of $L$ and $R$ by adding
all arcs from $L$ to $R$. Finally, we investigate whether, for all pairs of
positive integers $k_1,k_2$, there exists an integer $f(k_1,k_2)$ such that if
$D$ is an oriented graph with ${\rm inv}(D) \geq f(k_1,k_2)$ then there is a
partition $(V_1, V_2)$ of $V(D)$ such that ${\rm inv}(D\langle V_i\rangle) \geq
k_i$ for $i=1,2$. We show that $f(1,k)$ exists and $f(1,k)\leq k+10$ for all
positive integers $k$. Further, we show that $f(k_1,k_2)$ exists for all pairs
of positive integers $k_1,k_2$ when the oriented graphs in consideration are
restricted to be tournaments.
