Abstract:This paper contains a detailed discussion on the HOC(high order compact) method. The historyof the method is reviewed, the strongpoint of the method is pointed out,the basic properties, constructions and its present state are analyzed,together with the recent development. And especially, the mathematicalfoundations of the HOC-type method are discussed by summing up therelationship between the HOC and the rational approximation or Pad\'e-typeapproximations. Moreover, in the paper, rich information andrelated formulas are provided.