The aim of the paper under review is to develop a theory of triple covers in algebraic geometry. One of the most important general result obtained says that a triple cover \(X\to Y\) (with X and Y irreducible varieties over an algebraically closed field) is determined by a rank-two vector bundle E and a map \(S^ 3E\to \bigwedge^ 2E\), and conversely. Among the numerous corollaries of this theory we mention the following: (1) The general triple cover in dimension \(\geq 2\) has singular branch locus. (2) The general triple cover in dimension \(\geq 4\) is singular. (3) The moduli space of trigonal curves of genus \(g\) is connected, unirational, of dimension \(2g+1\). (4) An approach to construct surfaces of general type X with \(K^ 2\) arbitrarily close to 3e(X). Many other interesting results are also obtained.