Cognitive Informatics (CI) is a transdisciplinary enquiry of the internal information processing mechanisms and processes of the brain and natural intelligence shared by almost all science and engineering disciplines. This article presents an intensive review of the new field of CI. The structure of the theoretical framework of CI is described encompassing the Layered Reference Model of the Brain (LRMB), the OAR model of information representation, Natural Intelligence (NI) vs. Artificial Intelligence (AI), Autonomic Computing (AC) vs. imperative computing, CI laws of software, the mechanism of human perception processes, the cognitive processes of formal inferences, and the formal knowledge system. Three types of new structures of mathematics, Concept Algebra (CA), Real-Time Process Algebra (RTPA), and System Algebra (SA), are created to enable rigorous treatment of cognitive processes of the brain as well as knowledge representation and manipulation in a formal and coherent framework. A wide range of applications of CI in cognitive psychology, computing, knowledge engineering, and software engineering has been identified and discussed.