极小曲面是微分几何中研究最多的曲面之一，其理论已经发展成为微分几何的一个内容十分丰富的分支. 本文以微分几何理论为基础，从极小曲面定义出发，对欧氏空间中的极小曲面作综述. 本文由四个部分构成. 第一部分主要介绍了极小曲面的历史与发展现状，以及本文的主要工作；第二部分是本文的主要内容，首先以与极小曲面相关的微分几何的理论基础作为预备知识，作了相对较为详细的讲述，其次列举了三维欧氏空间中几个极小曲面的经典例子，并从微分几何理论的角度作了较为详细的说明，最后简要的列举了三维欧氏空间中极小曲面的共有的性质；第三部分则是力图将三维空间中曲面的微分几何相应的概念，推广到高维空间中的超曲面上，并给出高维欧氏空间中极小超曲面的定义，随后给出了高维欧氏空间中最基本的两种极小超曲面，并做了分析和说明；第四部分则是对全文工作的小结，以及还有待处理的问题.

Minimal Surface is one of the most important topics in differential geometry. In this thesis, we give a general survey about Minimal Surface in Euclidian Spaces. This thesis consists of four parts. The first part introduces the history and development of minimal surfaces, and the main work of the paper. The second part is the main content of this thesis. First, we introduce some basic knowledge about differential geometry and minimal surfaces. Secondly, we cited some classic examples of minimal surfaces in three-dimensional Euclidean space, and made some explanation from the view point of differential geometry. Finally, we list some general properties of minimal surfaces in the three-dimensional Euclidean space. The third part is trying to extend the theory of surfaces in three-dimensional space to the corresponding theory of hypersurfaces in high-dimensional space. The definition of minimal hypersurfaces is similar to that of in the three dimensional Euclidean space. Then I cite two basic minimal hypersurfaces in the high dimensional Euclidean space. The fourth part is a summary of the full text of the work, and the problems still to be processed.