This article is a part of the report for the research project ``Reform of the Course System and Teaching Content of Higher Mathematics (For Non-Mathematical Specialties)'' in 1995, supported by the National Ministry of Education. There are thirteen universities participated in this project. The Report not only reflects results of our participants, but also includes valuable opinions of many colleagues in the mathematical education circles. In this article, after a brief description on the history and reform situation of the higher mathematics education in China, attention concentrates to three aspects. They are: main problems in this field existing; the functions of mathematics accomplishment for college students; these concern course system, teaching and learning philosophy, such as overemphasized specialty education, overlooking to arouse rational thinking and aesthetic conceptions, etc. The last aspect contains a discussion on several important relationships, such as: knowledge impartment and quality cultivation, inherence and modernization of the mathematical knowledge, teacher's guidance and students' initiative, mathematical basic training and mathematical application consciousness and ability cultivation, etc.