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We establish the Noether inequality for projective $3$-folds. More precisely,we prove that the inequality $${\rm vol}(X)\geq\tfrac{4}{3}p_g(X)-{\tfrac{10}{3}}$$ holds for all projective $3$-folds $X$ ofgeneral type with either $p_g(X)\leq 4$ or $p_g(X)\geq 21$, where $p_g(X)$ isthe geometric genus and ${\rm vol}(X)$ is the canonical volume. This inequalityis optimal due to known examples found by M. Kobayashi in 1992.

The Noether inequality for algebraic threefolds (With an Appendix by J\'{a}nos Koll\'{a}r)