  This is the first in a series of papers concerned with Morse quasiflats,
which are a generalization of Morse quasigeodesics to arbitrary dimension. In
this paper we introduce a number of alternative definitions, and under
appropriate assumptions on the ambient space we show that they are equivalent
and quasi-isometry invariant; we also give a variety of examples. The second
paper proves that Morse quasiflats are asymptotically conical and have
canonically defined Tits boundaries; it also gives some first applications.
