Several papers (e.g. [7, 23, 42]) have recently sought to give general frameworks for online structures and algorithms ([4]), and seeking to connect, if only by analogy, online and computable structure theory. These initiatives build on earlier work on online colouring and other combinatorial algorithms by Bean [10], Kierstead, Trotter et al. [48, 54, 57] and others, as we discuss below. In this paper we will look at such frameworks and illustrate them with examples from the first author’s MSc Thesis (...) ([58]). In this thesis, Askes looks at online, adversarial online and strongly online algorithms and structures. Additionally, we prove some new theorems about online algorithms on graphs of bounded pathwidth as well as some new results on punctually 1-decidable structures. (shrink)