We consider the problem of computing a triangulation of the real projective plane $\mathbb{P}^2$, given a finite point set $\mathcal{P} = \{p_1, p_2,\ldots, p_n\}$ as input. We prove that a triangulation of $\mathbb{P}^2$ always exists if at least six points in $\mathcal{P}$ are in {\it general position}, i.e., no three of them are collinear. We also design a worst-case time $\mathcal{O}(n^2)$ algorithm for triangulating $\mathbb{P}^2$ if this necessary condition holds. We use the RAM model of computation for estimating the time complexity of our algorithm. As far as we know, this seems to be the first computational result on the real projective plane.