Diophantine equations are a rather diﬃcult topic. We may ask whether there exist solutions and, if so, whether there are inﬁnitely or only ﬁnitely many. In special examples answers to both questions were found (e.g., the Pell equation) but so far no general method for treating diophantine equations is known. With the ﬁniteness theorem 10.3 for the solutions to Thue equations in the previous chapter we have seen a tantalizing result with respect to the second question. However, the state of knowledge for another variety of diophantine equations is totally unclear. The most farreaching concept is the recent abc-conjecture which is the main theme of this chapter. Since its ﬁrst appearance in the 1980s, many applications to several quite diﬀerent problems were found — from diophantine equations via the class number problem in algebraic number theory to the prime number distribution. Besides, the abc-conjecture is a very general and rather simple statement which makes its application, saying it with the words of Granville & Tucker , as easy as abc.