  We propose the notion of integrable boundary in the context of discrete
integrable systems on quad-graphs. The equation characterizing the boundary
must satisfy a compatibility equation with the one characterizing the bulk that
we called the three-dimensional (3D) boundary consistency. In comparison to the
usual 3D consistency condition which is linked to a cube, our 3D boundary
consistency condition lives on a half of a rhombic dodecahedron. The We provide
a list of integrable boundaries associated to each quad-graph equation of the
classification obtained by Adler, Bobenko and Suris. Then, the use of the term
"integrable boundary" is justified by the facts that there are B\"acklund
transformations and a zero curvature representation for systems with boundary
satisfying our condition. We discuss the three-leg form of boundary equations,
obtain associated discrete Toda-type models with boundary and recover previous
results as particular cases. Finally, the connection between the 3D boundary
consistency and the set-theoretical reflection equation is established.
