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The purpose of this paper is to classify Anosov flows on the 3-manifoldsobtained by Dehn surgeries on the figure-eight knot. This set of 3-manifolds isdenoted by M(r) (r is a ratioanl number), which contains the first class ofhyperbolic 3-manifolds admitting Anosov flows in history, discovered byGoodman. Combining with the classification of Anosov flows on the sol-manifoldM(0) due to Plante, we have: 1. if r is an integer, up to topologicalequivalence, M(r) exactly carries a unique Anosov flow, which is constructed byGoodman by doing a Dehn-Fried-Goodman surgery on a suspension Anosov flow; 2.if r is not an integer, M(r) does not carry any Anosov flow. As a consequenceof the second result, we get infinitely many closed orientable hyperbolic3-manifolds which carry taut foliations but does not carry any Anosov flow. Thefundamental tool in the proofs is the set of branched surfaces built bySchwider, which is used to carry essential laminations on M(r).

Anosov flows on Dehn surgeries on the figure-eight knot