If \(X\) is a smooth projective variety of dimension \(n\) and \(L\) is a nef divisor on \(X\) such that \(aL - K_ X\) is nef and big for some \(a \geq 0\), then the base point free theorem says that the linear system \(| bD |\) is base point free for \(b \gg 0\). The author makes this statement effective by showing that the condition \(b \geq 2 (n + 2)!(a + n)\) is sufficient to guarantee base point freeness. The importance of this result is not the actual value of the bound but the fact that it depends only on \(n = \dim X\) and \(a\); a conjectured bound is \(b \geq a + n + 1\). The theorem is formulated in the more general case where \(X\) has log terminal singularities. The author applies his results to get explicit bounds for the number of irreducible families of \(n\)-dimensional smooth Fano varieties and polarized varieties.