  We investigate boundaries of 3D color codes and provide a systematic
classification into 101 distinct boundary types, including two novel classes.
The first class consists of 1 boundary and is generated by sweeping the
codimension-1 (2D) $T$-domain wall across the system and attaching it to the
$X$-boundary that condenses only magnetic fluxes. Since the $T$-domain wall
cannot condense on the $X$-boundary, a new magic boundary is produced, where
the boundary stabilizers contain $XS$-stabilizers going beyond the conventional
Pauli stabilizer formalism, and hence contains 'magic'. Neither electric nor
magnetic excitations can condense on such a magic boundary, and only the
composite of the magnetic flux and codimension-2 (1D) $S$-domain wall can
condense on it, which makes the magic boundary going beyond the classification
of the Lagrangian subgroup. The second class consists of 70 boundaries and is
generated by sweeping the $S$-domain wall across a codimension-1 submanifold
and attaching it to the boundary. This generates a codimension-2 (1D) nested
boundary at the intersection. We also connect these novel boundaries to their
previously discovered counterpart in the $\mathbb{Z}_2^3$ gauge theory,
equivalent to three copies of 3D toric codes, where the $S$ and $T$ domain
walls correspond to gauged symmetry-protected topological (SPT) defects. New
boundaries are produced whenever the corresponding symmetry of the SPT defect
remains unbroken on the boundary. Applications of the magic boundaries include
implementing fault-tolerant non-Clifford logical gates, e.g., in the context of
fractal topological codes.
