Let $X$ be a normal projective Fano variety, that is the anti-canonical divisor $-K\_X$ is ample. 


For any $m>0$ let us consider the complete linear system $|-mK\_X|$ and the map
$$f\_{|-mK\_X|}:X\dashrightarrow X\_m\subseteq\mathbb{P}(|-mK\_X|) = \mathbb{P}^{N\_m}$$
where $X\_m$ is the closure of $f\_{|-mK\_X|}(X)$ in $\mathbb{P}^{N\_m}$.


Do there exist estimates for the smallest rational number $m>0$ such that $f\_{|-mK\_X|}:X\dashrightarrow X\_m$ is birational ? 



Anti-canonical divisor of a Fano variety
If you want to consider smooth (weak) Fano variety, then Fukuda has effective estimation of the birationality of anti-canonical systems for any dimension (but not optimal), see [S. FUKUDA, A note on Ando’s paper “pluricanonical systems of algebraic varieties of general type of dimension ≤ 5, Tokyo J. Math., 14(1991) 479-487.]


If you want to consider Fano variety with at worst canonical singularities, then we know the answer up to dimension 3, see my joint work with Meng Chen, [On the anti-canonical geometry of Q-fano threefolds, J. Differential Geom. 104 (2016), no.1, 59--109]


If you want to consider Fano varieties with klt or lc singularities, there is so far no explicit answer to your question even in dimension 2. 


Of course I should mention that the existence of such effective birationality for $\epsilon$-lc Fano is proved by Birkar last year, but his method does not give a way to estimate the number $m$. 


By the way, if you are interested in Fano varieties with klt singularities which is K-semistable (e.g. admitting Kahler--Einstein metrics), then in this case we have effective estimation of $m$ by my recent work, see proof of Theorem 1.5 of arXiv:1705.02740.


