  The color code is both an interesting example of an exactly solved
topologically ordered phase of matter and also among the most promising
candidate models to realize fault-tolerant quantum computation with minimal
resource overhead. The contributions of this work are threefold. First of all,
we build upon the abstract theory of boundaries and domain walls of topological
phases of matter to comprehensively catalog the objects realizable in color
codes. Together with our classification we also provide lattice representations
of these objects which include three new types of boundaries as well as a
generating set for all 72 color code twist defects. Our work thus provides an
explicit toy model that will help to better understand the abstract theory of
domain walls. Secondly, we discover a number of interesting new applications of
the cataloged objects for quantum information protocols. These include improved
methods for performing quantum computations by code deformation, a new
four-qubit error-detecting code, as well as families of new quantum
error-correcting codes we call stellated color codes, which encode logical
qubits at the same distance as the next best color code, but using
approximately half the number of physical qubits. To the best of our knowledge,
our new topological codes have the highest encoding rate of local stabilizer
codes with bounded-weight stabilizers in two dimensions. Finally, we show how
the boundaries and twist defects of the color code are represented by multiple
copies of other phases. Indeed, in addition to the well studied comparison
between the color code and two copies of the surface code, we also compare the
color code to two copies of the three-fermion model. In particular, we find
that this analogy offers a very clear lens through which we can view the
symmetries of the color code which gives rise to its multitude of domain walls.
