This article discusses the concept and applications of cognitive dynamic systems (CDS), which are a type of intelligent system inspired by the brain. There are two branches of CDS, one for linear and Gaussian environments (LGEs), such as cognitive radio and cognitive radar, and another one for non-Gaussian and nonlinear environments (NGNLEs), such as cyber processing in smart systems. Both branches use the same principle, called the perception action cycle (PAC), to make decisions. The focus of this review is on the applications of CDS, including cognitive radios, cognitive radar, cognitive control, cyber security, self-driving cars, and smart grids for LGEs. For NGNLEs, the article reviews the use of CDS in smart e-healthcare applications and software-defined optical communication systems (SDOCS), such as smart fiber optic links. The results of implementing CDS in these systems are very promising, with improved accuracy, performance, and lower computational costs. For example, CDS implementation in cognitive radars achieved a range estimation error that is as good as 0.47 (m) and a velocity estimation error of 3.30 (m/s), outperforming traditional active radars. Similarly, CDS implementation in smart fiber optic links improved the quality factor by 7 dB and the maximum achievable data rate by 43% compared to those of other mitigation techniques.