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Let $V$ be a complex nonsingular projective 3-fold of general type. We prove$P_{12}(V)>0$ and $P_{24}(V)>1$ (which answers an open problem of J. Kollar andS. Mori). We also prove that the canonical volume has an universal lower bound$\text{Vol}(V) \geq 1/2660$ and that the pluri-canonical map $\Phi_m$ isbirational onto its image for all $m\geq 77$. As an application of our method,we prove Fletcher's conjecture on weighted hyper-surface 3-folds with terminalquotient singularities. Another featured result is the optimal lower bound$\text{Vol}(V)\geq {1/420}$ among all those 3-folds $V$ with $\chi({\mathcalO}_V)\leq 1$.

Explicit birational geometry of threefolds of general type