Learning-based　image　coding　has　attracted　increasing　attentions　for　its　higher　compression　efficiency　than　reigning　image　codecs.　However,　most　existing　learning-based　codecs　do　not　support　variable　rates　with　a　single　encoder;　their　decoders　are　also　of　fixed,　high　computational　complexity.　In　this　paper,　we　propose　an　End-to-end,　Learning-based　and　Flexible　Image　Codec　(ELFIC)　that　supports　variable　rate　and　flexible　decoding　complexity.　First,　we　propose　a　general　image　codec　with　Nonlinear　Feature　Fusion　Transform　(NFFT)　as　nonlinear　transforms　to　improve　its　Rate-Distortion　(RD)　performance.　Second,　we　propose　an　Instance-aware　Decoding　Complexity　Allocation　(IDCA)　approach,　which　exploits　image　contents　for　a　tradeoff　between　reconstruction　quality　and　computational　complexity　in　the　decoding　process.　Third,　we　propose　an　RD-Complexity　(RDC)　optimization　algorithm,　which　maximizes　the　image　quality　under　given　rate　and　complexity　constraints　for　the　whole　framework.　Experimental　results　show　that　ELFIC　achie-ves　variable　rate,　flexible　decoding　complexity　with　the　state-of-the-art　RD　performance.　It　also　supports　a　more　efficient　decoding　process　by　focusing　on　image　contents.　Source　codes　are　available　at　https://github.com/Zhichen-Zhang/ELFIC-Image-Compression.　©　2023　ACM.