  We answer a number of questions of Erd\H{o}s on the existence of arithmetic
progressions in $k$-full numbers (i.e. integers with the property that every
prime divisor necessarily occurs to at least the $k$-th power). Further, we
deduce a variety of arithmetic constraints upon such progressions, under the
assumption of the $abc$-conjecture of Masser and Oesterl\'e.
