Fra\"iss\'e limits of metric structures

We develop \emph{Fra"iss'e theory}, namely the theory of \emph{Fra"iss'e classes} and \emph{Fra"iss'e limits}, in the context of metric structures. We show that a class of finitely generated structures is Fra"iss'e if and only if it is the age of a separable approximately homogeneous structure, and conversely, that this structure is necessarily the unique limit of the class, and is universal for it. We do this in a somewhat new approach, in which ‘‘finite maps up to errors’’ are coded by \emph{approximate isometries}.