We say that a mapping $f$ from a Banach space $X$ to another Banach space $Y$ is an $\varepsilon$-isometry for some $\varepsilon\geq 0$ provided $$\Big|\|f(x)-f(y)\|-\|x-y\|\Big|\leq\varepsilon, \;\;\forall x,y\in X;$$ A $0$-isometry  is simply called an isometry. The study of isometry and its generalization $\varepsilon$-isometry has continued for over 80 years since Mazur and Ulam's celebrated theorem (1932): Every surjective isometry $f:X\rightarrow Y$ is necessarily affine. In this talk, after introducing related concepts of isometry and $\varepsilon$-isometry, we give examples showing that these concepts have wide backgrounds in pure mathematics and applied mathematics. Then we present a brief survey on this topic, especially. introduce the recent development. Finally, we conclude this talk by proposing some open problems on this research area.

程立新，厦门大学数学科学学院教授、博士生导师，厦门大学陈景润特聘教授。 2002年入选"教育部跨世纪优秀人才培养计划" 。现任全国泛函分析空间理论学术联络组成员, 全国非线性泛函分析大会联络组成员, 全国空间理论和应用泛函分析大会副主席, 曾担任国家自然科学基金委员会数理科学部特邀评委和中国数学会常务理事以及福建省数学会理事长。学术期刊《数学进展》、《应用泛函分析学报》、 《数学研究》、《数学研究期刊 《International Journal of Functional Analysis, Operator Theory and Applications 》、 和《The Open Mathematics Journal》编委,《厦门大学学报》副主编，代表性论文发表在《J. Funct. Anal.》、《Israel J. Math》、《Studia Math.》等著名国际期刊上。