The　connectivity　of　a　graph　G,　κ(G),　is　the　minimum　number　of　vertices　whose　removal　disconnects　G,　and　the　value　of　κ(G)　can　be　determined　using　Menger＇s　theorem.　It　has　long　been　one　of　the　most　important　factors　that　characterize　both　graph　reliability　and　fault　tolerability.　Two　extensions　to　the　classic　notion　of　connectivity　were　introduced　recently:　structure　connectivity　and　substructure　connectivity.　Let　H　be　isomorphic　to　any　connected　subgraph　of　G.　The　H-structure　connectivity　of　G,　denoted　by　κ(G;　H),　is　the　cardinality　of　a　minimum　set　F　of　connected　subgraphs　in　G　such　that　every　element　of　F　is　isomorphic　to　H,　and　the　removal　of　F　disconnects　G.　The　H-substructure　connectivity　of　G,　denoted　by　κ(G;　H),　is　the　cardinality　of　a　minimum　set　X　of　connected　subgraphs　in　G　whose　removal　disconnects　G　and　every　element　of　X　is　isomorphic　to　a　connected　subgraph　of　H.　The　family　of　hypercube-like　networks　includes　many　well-defined　network　architectures,　such　as　hypercubes,　crossed　cubes,　twisted　cubes,　and　so　on.　In　this　paper,　both　the　structure　and　substructure　connectivity　of　hypercube-like　networks　are　studied　with　respect　to　the　m-star　K1,m　structure,　m　≥　1,　and　the　4-cycle　C4　structure.　Moreover,　we　consider　the　relationships　between　these　parameters　and　other　concepts.　©　2020　World　Scientific　Publishing　Company.