  We present an overview of the study of the Thurston norm, introduced by W. P.
Thurston in the seminal paper "A norm for the homology of 3-manifolds" (written
in 1976 and published in 1986). We first review fundamental properties of the
Thurston norm of a 3-manifold, including a construction of codimension-1 taut
foliations from norm-minimizing embedded surfaces, established by D. Gabai. In
the main part we describe relationships between the Thurston norm and other
topological invariants of a 3-manifold: the Alexander polynomial and its
various generalizations, Reidemeister torsion, the Seiberg-Witten invariant,
Heegaard Floer homology, the complexity of triangulations and the profinite
completion of the fundamental group. Some conjectures and questions on related
topics are also collected. The final version of this paper will appear as a
chapter in the book "In the tradition of Thurston, II", edited by K. Ohshika
and A. Papadopoulos (Springer, 2022).
