Fractal　image　compression　is　a　relatively　recent　image　compression　method.　Although　it　does　not　work　as　good　as　the　state-of-the-art　compression　technology,　its　main　advantage　that　the　decompressing　algorithm　is　very　simple,　makes　it　suitable　for　the　situation　of　one　encoding　and　many　decoding,　such　as　video　on　demand,　archive　compression,　etc.　There　are　two　basic　fractal　compression　methods,　namely　the　cube-based　and　the　frame-based　methods,　being　commonly　used　in　the　industry.　However　there　are　advantages　and　disadvantages　in　both　methods.　This　paper　discusses　the　two　basic　fractal　and　other　algorithms　extend　from　them.　Experimental　results　show　that　the　algorithm　based　on　adaptive　partition　can　obtain　a　much　higher　compression　ratio　compared　to　the　algorithm　based　on　fixed　partition　while　the　qualities　of　decompressed　images　are　similar,　and　the　hybrid　algorithm　improves　the　compression　ratio　and　the　quality　of　decompressed　images.