  We introduce a new type of boundary for proper geodesic spaces, called the
Morse boundary, that is constructed with rays that identify the "hyperbolic
directions" in that space. This boundary is a quasi-isometry invariant and thus
produces a well-defined boundary for any finitely generated group. In the case
of a proper $\mathrm{CAT}(0)$ space this boundary is the contracting boundary
of Charney and Sultan and in the case of a proper Gromov hyperbolic space this
boundary is the Gromov boundary. We prove three results about the Morse
boundary of Teichm\"uller space. First, we show that the Morse boundary of the
mapping class group of a surface is homeomorphic to the Morse boundary of the
Teichm\"uller space of that surface. Second, using a result of Leininger and
Schleimer, we show that Morse boundaries of Teichm\"uller space can contain
spheres of arbitrarily high dimension. Finally, we show that there is an
injective continuous map of the Morse boundary of Teichm\"uller space into the
Thurston compactification of Teichm\"uller space by projective measured
foliations.
  An appendix includes a corrigendum to the paper introducing refined Morse
gauges to correct the proof and statement of Lemma 2.10.
