Informally, a set operad is a family of combinatorial structures plus 1) an (associative) mechanism that creates larger structures from smaller using as assembler an external structure in the same family 2) Identity structures over singleton sets. We give the precise definition of set operads in the context of Joyal’s combinatorial species. Interesting examples of set operads are Graphs, directed graphs, posets, Boolean functions, monotone Boolean functions, simple multiperson games, simplicial complexes, etc. We show that Decompositions theory of combinatorial structure can be formulated in the general context of set operads by using the operation of amalgam (coproduct) of operads.