This is a short introductory review on application of quantum probability (QP) in physics, cognitive studies, artificial intelligence, psychology, decision making, social and political sciences. We emphasize that QP is a contextual probability theory that is based on the quantum-like contextual paradigm: QP formalism can be used to model behavior not only of genuine quantum physical systems, but all context-sensitive systems, e.g., human beings. We especially are interested in the quantum-like modeling of rationality (cf. with classical approach to rationality which is based on the Bayesian update). Quantum rationality is introduced as decision making via quantum information processing. Quantum-like and classical rational agents behaves in very different ways. For example, a quantum-like agent can violate the Savage Sure Thing Principle and the Aumann theorem on impossibility of agreeing to disagree. Irrational (from the classical viewpoint) behavior can be profitable from the information processing viewpoint, especially for agents who are overloaded by information flows. Quantum-like agents can save a lot of information processing resources. At the same time, this sort of rationality make humans a convenient active medium for socio-political engineering, e.g., social lasing - Stimulated Amplification of Social Actions (SASA). This rationality plays the important role in the process of decision making not only by biosystems, but even by AI-systems. The latter equipped with quantum(-like) information processors would behave irrationally, from the classical viewpoint. As for biosystems, quantum rational behavior of AI-systems has its advantages and disadvantages. Quantum-like information processing in AI-systems can be based on classical physical devices, e.g., classical digital or analog computers.