  We consider theories consisting of a planar interface with $\N=4$
super-Yang-Mills on either side and varying gauge coupling across the
interface. The interface does not carry any independent degrees of freedom, but
is allowed to support local gauge invariant operators, included with
independent interface couplings. In general, both conformal symmetry and
supersymmetry will be broken, but for special arrangements of the interface
couplings, these symmetries may be restored. We provide a systematic
classification of all allowed interface supersymmetries. We find new theories
preserving eight and four Poincar\'e supersymmetries, which get extended to
sixteen and eight supersymmetries in the conformal limit, respectively with
$SU(2) \times SU(2)$, $SO(2) \times SU(2)$ internal symmetry. The Lagrangians
for these theories are explicitly constructe d. We also recover the theory with
two Poincar\'e supersymmetries and SU(3) internal symmetry proposed earlier as
a candidate CFT dual to super Janus. Since our new interface theories have only
operators from the supergravity multiplet turned on, dual supergravity
solutions are expected to exist. We speculate on the possible relation between
the interface theory with maximal supersymmetry and the near-horizon limit of
the D3-D5 system.
