  We study the operator algebra of extended conformal defects in more than two
spacetime dimensions. Such algebra structure encodes the combined effect of
multiple impurities on physical observables at long distances as well as the
interactions among the impurities. These features are formalized by a fusion
product which we define for a pair of defects, after isolating divergences that
capture the effective potential between the defects, which generalizes the
usual Casimir energy. We discuss general properties of the corresponding fusion
algebra and contrast with the more familiar cases that involve topological
defects. We also describe the relation to a different defect setup in the shape
of a wedge. We provide explicit examples to illustrate these properties using
line defects and interfaces in the Wilson-Fisher CFT and the
Gross-Neveu(-Yukawa) CFT and determine the defect fusion data thereof.
