Because　of　their　unique　physical　properties,　two-dimensional　(2D)　materials　have　attracted　extensive　interest　as　potential　candidates　for　next　generation　electrical　and　optoelectronic　applications;　in　particular,　large　optical　second　harmonic　generation　(SHG)　has　been　recently　discovered　in　a　number　of　2D　materials,　which　allows　them　to　have　valuable　applications　in　electro-optic　modulators　and　switches,　frequency　conversion,　and　so　forth.　Here,　we　performed　a　detailed　first-principles　study　on　the　geometries,　electronic　structures　and　SHG　properties　of　a　recently　prepared　In2Se3　monolayer.　Our　results　reveal　that　In2Se3　with　a　single-layered　structure　may　be　a　material　with　the　strongest　SHG　response　among　IIIA-VIA　semiconductors　reported　to　date,　and　the　maximum　magnitude　of　the　static　SHG　coefficient　is　predicted　to　be　208.8　pm　V-1.　The　extraordinary　SHG　is　strongly　correlated　with　a　special　arrangement　of　two　kinds　of　In-Se　polyhedra,　namely,　[InSe6]9-　octahedra　and　[InSe4]5-　tetrahedra,　on　the　opposite　sides　of　the　In2Se3　monolayer.　Further　investigations　on　other　In-Se　compounds　with　different　structures　and　stoichiometries　indicate　that　besides　the　configuration　of　the　In-Se　building　block,　the　SHG　response　can　be　significantly　influenced　by　the　interlayer　interactions,　and　the　magnitude　of　the　SHG　coefficient　of　the　In2Se3　monolayer　decreases　gradually　as　the　interlayer　distance　decreases　until　it　approaches　that　of　In2Se3　bulk　in　the　R3m　phase.　Our　present　findings　provide　a　deep　understanding　of　the　SHG　properties　in　IIIA-VIA　2D　semiconductors　and　are　also　helpful　for　designing　new　2D　materials　with　tunable　SHG　susceptibility.　©　2018　The　Royal　Society　of　Chemistry.