We study the complexifications of real operator spaces.
We show that for every real operator space $V$ there exists a
unique complex operator space matrix norm $\{\|\cdot\|_n\}$ on its
complexification $V_c = V \+{\rm i} V$ which extends the original
matrix norm on $V$ and satisfies the condition
$\|x +{\rm i}y\|_n = \|x -{\rm i}y\|_n$ for all
$x + {\rm i} y \in M_n(V_c) = M_n(V) \+ {\rm i} M_n(V)$.
As a consequence of this
result, we characterize complex operator spaces
which can be expressed as the complexification of some real operator
space. Finally, we show that some properties of real operator spaces
are closely related to the corresponding properties of their
complexifications.