Macpherson and Steinhorn developed the notion of 1-dimensional asymptotic classes, which are classes of finite structures with notions of measure and dimension coming from the study of the size of their definable sets. The seminal example of these classes is the class of finite fields, thanks to a remarkable result of Chatzidakis, Macintyre and Van den Dries. In this talk I will review the main definitions and results of 1-dimensional asymptotic classes, and present the concept of o–asymptotic classes, which is intended to be an adaptation of the 1-dimensional asymptotic classes to the context of finite linearly ordered structures. I will also present a classification of the ultraproducts of o-asymptotic classes, showing that they are superrosy of U-thorn rank 1 and NTP2 of burden 1, and discuss some examples and non-examples.