Computable presentability of countable linear orders

The main goal of this paper is to study algorithmic properties of countable linear orders by constructing effective presentations of these structures on the set of natural numbers. In 1991, C. Jockusch and R. Soare constructed a low linear order without computable presentations. Before in 1989, R. Downey and M. Moses showed that each low discrete linear order has a computable copy. It is natural to ask for which order types of low presentations it is sufficient the existence of a computable presentation. This question (namely, research program) was stated by R. Downey in 1998: Describe the order property $P$ such that, for any low linear order $L$, $P(L)$ implies the existence of a computable presentation of $L$. In this paper, we give a detailed review of the main results in this direction. These results are mostly obtained by the author or in co-authorship.