Anti-pluricanonical systems on Fano varieties

In this paper, we study the linear systems $|-mK_X|$ on Fano varieties $X$ with klt singularities. In a given dimension $d$, we prove $|-mK_X|$ is non-empty and contains an element with "good singularities" for some natural number $m$ depending only on $d$; if in addition $X$ is $ε$-lc for some $ε>0$, then we show that we can choose $m$ depending only on $d$ and $ε$ so that $|-mK_X|$ defines a birational map. Further, we prove Shokurov's conjecture on boundedness of complements, and show that certain classes of Fano varieties form bounded families.