  Let a ``complex probability'' be a normalizable complex distribution $P(x)$
defined on $\R^D$. A real and positive probability distribution $p(z)$, defined
on the complex plane $\C^D$, is said to be a positive representation of $P(x)$
if $\langle Q(x)\rangle_P = \langle Q(z)\rangle_p$, where $Q(x)$ is any
polynomial in $\R^D$ and $Q(z)$ its analytical extension on $\C^D$. In this
paper it is shown that every complex probability admits a real representation
and a constructive method is given. Among other results, explicit positive
representations, in any number of dimensions, are given for any complex
distribution of the form Gaussian times polynomial, for any complex
distributions with support at one point and for any periodic Gaussian times
polynomial.
