Anti-pluricanonical systems on Fano varieties

In this paper, we study the linear systems |-mKX| on Fano varieties X with klt singularities. In a given dimension d, we prove |-mKX| 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 |-mKX| 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.