Conjugated microporous polymers (CMPs) are a class of materials with unique structural properties, including extended π-conjugation and permanent microporosity, with a huge synthetic diversity offering up a number of topological strategies for control of their amorphous structure and properties. This provides a platform for the exploration of chemical and electronic structure properties that is not available for other classes of materials. CMPs have applications in gas storage, gas separation, heterogeneous catalysis, chemosensors, light harvesting devices, polymer light emitting diodes and as supercapacitors. There is great potential for as yet undiscovered applications as we further explore their synthetic diversity and gain new strategies for controlling structure. In this chapter, we define the core chemical and structural properties of CMPs and the synthetic strategies adopted. We discuss the various methods of analysing and rationalising the CMP molecular structure and porous properties, the particular challenges in elucidating the structure of amorphous CMPs and strategies for tackling these challenges. Finally, we discuss limitations for CMP materials and future directions that might overcome these challenges and open up new areas for exploration.