In mathematics, we know there are some concepts - objects, constructions, structures, proofs - that are more complex and difficult
to describe than others. Computable structure theory quantifies and studies the complexity of mathematical structures, structures
such as graphs, groups, and orderings. Written by a contemporary expert in the subject, this is the first full monograph on
computable structure theory in 20 years. Aimed at graduate students and researchers in mathematical logic, it brings new results
of the author together with many older results that were previously scattered across the literature and presents them all
in a coherent framework, making it easier for the reader to learn the main results and techniques in the area for application
in their own research. This volume focuses on countable structures whose complexity can be measured within arithmetic; a forthcoming
second volume will study structures beyond arithmetic.