The　processor　failures　in　a　multiprocessor　system　have　a　negative　impact　on　its　distributed　computing　efficiency.　Because　of　the　rapid　expansion　of　multiprocessor　systems,　the　importance　of　fault　diagnosis　is　becoming　increasingly　prominent.　The　h-component　diagnosability　of　G,　denoted　by　ct(h)(G),　is　the　maximum　number　of　nodes　of　the　faulty　set　F　that　is　correctly　identified　in　a　system,　and　the　number　of　components　in　G　-　F　is　at　least　h.　In　this　paper,　we　determine　the　(h　+　1)-component　diagnosability　of　general　networks　under　the　PMC　model　and　MM*　model.　As　applications,　the　component　diagnosability　is　explored　for　some　well-known　networks,　including　complete　cubic　networks,　hierarchical　cubic　networks,　generalized　exchanged　hypercubes,　dual-cube-like　networks,　hierarchical　hypercubes,　Cayley　graphs　generated　by　transposition　trees　(except　star　graphs),　and　DQcube　as　well.　Furthermore,　we　provide　some　comparison　results　between　the　component　diagnosability　and　other　fault　diagnosabilities.