  We study the notion of an asymptotically automatic sequence, which
generalises the notion of an automatic sequence. While $k$-automatic sequences
are characterised by finiteness of $k$-kernels, the $k$-kernels of
asymptotically $k$-automatic sequences are only required to be finite up to
equality almost everywhere. We prove basic closure properties and a linear
bound on asymptotic subword complexity, show that results concerning
frequencies of symbols are no longer true for the asymptotic analogue, and
discuss some classification problems.
