  The abc conjecture, one of the most famous open problems in number theory,
claims that three positive integers satisfying a+b=c cannot simultaneously have
significant repetition among their prime factors; in particular, the product of
the distinct primes dividing the three integers should never be much less than
c. Triples of numbers satisfying a+b=c are called abc triples if the product of
their distinct prime divisors is strictly less than c. We catalog what is known
about abc triples, both numerical examples found through computation and
infinite familes of examples established theoretically. In addition, we collect
motivations and heuristics supporting the $abc$ conjecture, as well as some of
its refinements and generalizations, and we describe the state-of-the-art
progress towards establishing the conjecture.
