an *$abc$-triple* is a triple of pairwise coprime positive integers $a$, $b$, $c$ with $a$ + $b$ = $c$. the *radical* $r$ of such a triple is the product of the distinct prime numbers dividing $abc$, and the *quality* $q$ is defined by $q = (log c)/log r$. for example, the triple given by $a = 5, b = 27, c = 32$ has $r = 30$ and $q = (log 32)/log 30 = 1.018975235$... the *$abc$-conjecture* asserts that for any real number $q > 1$, the number of $abc$-triples with quality greater than $q$ is finite. it is known that there do exist infinitely many $abc$-triples with quality greater than $1$. the main subject of the lecture is an algorithm for listing, given a large integer $n$, all $abc$-triples with $c$ at most $n$ and quality greater than $1$. as a byproduct, the algorithm yields an upper bound for the number of such triples, as a function of $n$.