Explore computable structure theory and its applications to noncomputable structures in this 56-minute Association for Symbolic Logic Invited Address. Delve into the basics of categoricity, computability, and effective categoricity. Examine alternative paths and definitions of computable categoricity for computable structures. Investigate hands-on examples, including W-computably categorical fields and noncomputable fields. Analyze the number of structures, isomorphism types of subfields, and their relationships to computable fields. Gain insights into theorems related to Rx and HTP(K), and uncover answers to complex mathematical questions in this comprehensive talk given by Russell Miller from Queen's College and CUNY Graduate Center at the Joint Mathematics Meetings.