Our adventure with corings starts here. Corings should be seen as one of the most fundamental algebraic structures that include rings and coalgebras as special cases. They appear naturally in the following chain of generalisations: coalgebras over fields; coalgebras over commutative rings; and coalgebras over noncommutative rings (= corings). The scope of applications of corings is truly amazing, and their importance can be explained at least on two levels. First, corings are a “mild” generalisation of coalgebras, in the sense that several properties of coalgebras over commutative rings carry over to corings. In particular, various general properties of corings can be proven by using the same techniques as for coalgebras over rings. From this point of view the step from coalgebras over fields to coalgebras over commutative rings is much bolder and adventurous than that from coalgebras over rings to corings. Second, the range of problems that can be described with the use of corings is much wider than the problems that could ever be addressed by coalgebras. In situations such as ring extensions, even if a commutative ring is extended, one will always require corings. In addition to all this, the theory of corings also has an extremely useful unifying power. We shall soon see that several results about the structure of Hopf modules (cf. Section 14), including the Fundamental Theorem of Hopf algebras 15.5, and their generalisations to different classes of Hopf-type modules, are simply special cases of structure theorems for corings.