The chapter introduces the concept of statistical metric spaces, which generalize the traditional metric spaces by associating a distribution function with each pair of elements instead of a single real number. This generalization is motivated by the fact that in many practical applications, a single measurement often does not provide a precise distance but rather a range of possible values. The chapter outlines the historical development of statistical metric spaces, starting with Menger's introduction in 1942, followed by Wald's critique and subsequent improvements. It also discusses the axiomatics of statistical metric spaces, focusing on the triangle inequality, and explores specific types of spaces such as equilateral and simple spaces. The chapter concludes with a discussion on topological notions and the continuity properties of the distance function in statistical metric spaces.The chapter introduces the concept of statistical metric spaces, which generalize the traditional metric spaces by associating a distribution function with each pair of elements instead of a single real number. This generalization is motivated by the fact that in many practical applications, a single measurement often does not provide a precise distance but rather a range of possible values. The chapter outlines the historical development of statistical metric spaces, starting with Menger's introduction in 1942, followed by Wald's critique and subsequent improvements. It also discusses the axiomatics of statistical metric spaces, focusing on the triangle inequality, and explores specific types of spaces such as equilateral and simple spaces. The chapter concludes with a discussion on topological notions and the continuity properties of the distance function in statistical metric spaces.