This survey provides an overview of the concordance group of knots in three-dimensional space. It begins with a review of the definitions of knots and concordance and then presents the aspects of the algebraic theory of concordance (integral invariants, signatures, the Arf invariant: \(\mathbb Z_{2}\), Polynomial invariants: \(\mathbb Z_{2}\), \(W(\mathbb Q):\mathbb Z_{2}\) and \(\mathbb Z_{4}\) invariants). There are two related theories of concordance, one in the smooth category and the other topological.
The author focuses on the smooth setting, though differences and main results in the topological setting are included. Casson-Gordon invariants are examined in detail (construction of companions, genus one knots and the Seifert form). Recent results from the topological locally flat category are presented, as are new applications from smooth geometry. A discussion of the interplay between 3-dimensional knot properties and concordance is presented (primeness, knot symmetry: amphicheirality, reversibility and mutation, periodicity, genus, fibering, unknotting number). The survey concludes with a brief list of open problems and an extensive bibliography.
For the entire collection see [Zbl 1073.57001].