Separable　nonlinear　least　squares　(SNLLS)　problems　have　attracted　interest　in　a　wide　range　of　research　fields　such　as　machine　learning,　computer　vision,　and　signal　processing.　During　the　past　few　decades,　several　algorithms,　including　the　joint　optimization　algorithm,　alternated　least　squares　(ALS)　algorithm,　embedded　point　iterations　(EPI)　algorithm,　and　variable　projection　(VP)　algorithms,　have　been　employed　for　solving　SNLLS　problems　in　the　literature.　The　VP　approach　has　been　proven　to　be　quite　valuable　for　SNLLS　problems　and　the　EPI　method　has　been　successful　in　solving　many　computer　vision　tasks.　However,　no　clear　explanations　about　the　intrinsic　relationships　of　these　algorithms　have　been　provided　in　the　literature.　In　this　paper,　we　give　some　insights　into　these　algorithms　for　SNLLS　problems.　We　derive　the　relationships　among　different　forms　of　the　VP　algorithms,　EPI　algorithm　and　ALS　algorithm.　In　addition,　the　convergence　and　robustness　of　some　algorithms　are　investigated.　Moreover,　the　analysis　of　the　VP　algorithm　generates　a　negative　answer　to　Kaufman＇s　conjecture.　Numerical　experiments　on　the　image　restoration　task,　fitting　the　time　series　data　using　the　radial　basis　function　network　based　autoregressive　(RBF-AR)　model,　and　bundle　adjustment　are　given　to　compare　the　performance　of　different　algorithms.