This　paper　offers　an　extensive　overview　of　the　utilization　of　sequential　approximate　optimization　approaches　in　the　context　of　numerically　simulated　large-scale　continuum　structures.　These　structures,　commonly　encountered　in　engineering　applications,　often　involve　complex　objective　and　constraint　functions　that　cannot　be　readily　expressed　as　explicit　functions　of　the　design　variables.　As　a　result,　sequential　approximation　techniques　have　emerged　as　the　preferred　strategy　for　addressing　a　wide　array　of　topology　optimization　challenges.　Over　the　past　several　decades,　topology　optimization　methods　have　been　advanced　remarkably　and　successfully　applied　to　solve　engineering　problems　incorporating　diverse　physical　backgrounds.　In　comparison　to　the　large-scale　equation　solution,　sensitivity　analysis,　graphics　post-processing,　etc.,　the　progress　of　the　sequential　approximation　functions　and　their　corresponding　optimizers　make　sluggish　progress.　Researchers,　particularly　novices,　pay　special　attention　to　their　difficulties　with　a　particular　problem.　Thus,　this　paper　provides　an　overview　of　sequential　approximation　functions,　related　literature　on　topology　optimization　methods,　and　their　applications.　Starting　from　optimality　criteria　and　sequential　linear　programming,　the　other　sequential　approximate　optimizations　are　introduced　by　employing　Taylor　expansion　and　intervening　variables.　In　addition,　recent　advancements　have　led　to　the　emergence　of　approaches　such　as　Augmented　Lagrange,　sequential　approximate　integer,　and　non　-gradient　approximation　are　also　introduced.　By　highlighting　real　-world　applications　and　case　studies,　the　paper　not　only　demonstrates　the　practical　relevance　of　these　methods　but　also　underscores　the　need　for　continued　exploration　in　this　area.　Furthermore,　to　provide　a　comprehensive　overview,　this　paper　offers　several　novel　developments　that　aim　to　illuminate　potential　directions　for　future　research.