  Defects between gapped boundaries provide a possible physical realization of
projective non-abelian braid statistics. A notable example is the projective
Majorana/parafermion braid statistics of boundary defects in fractional quantum
Hall/topological insulator and superconductor heterostructures. In this paper,
we develop general theories to analyze the topological properties and
projective braiding of boundary defects of topological phases of matter in two
spatial dimensions. We present commuting Hamiltonians to realize defects
between gapped boundaries in any $(2+1)D$ untwisted Dijkgraaf-Witten theory,
and use these to describe their topological properties such as their quantum
dimension. By modeling the algebraic structure of boundary defects through
multi-fusion categories, we establish a bulk-edge correspondence between
certain boundary defects and symmetry defects in the bulk. Even though it is
not clear how to physically braid the defects, this correspondence elucidates
the projective braid statistics for many classes of boundary defects, both
amongst themselves and with bulk anyons. Specifically, three such classes of
importance to condensed matter physics/topological quantum computation are
studied in detail: (1) A boundary defect version of Majorana and parafermion
zero modes, (2) a similar version of genons in bilayer theories, and (3)
boundary defects in $\mathfrak{D}(S_3)$.
