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Cognitive autonomy of an autonomous system depends on its cyber module's ability to comprehend the actions and intent of the applications and services running on that system. The autonomous system should be able to accomplish this without or with limited human intervention. These mission-critical autonomous systems are often deployed in unpredictable and dynamic environments and are vulnerable to evasive cyberattacks. In particular, some of these cyberattacks are Advanced Persistent Threats where an attacker conducts reconnaissance for a long period time to ascertain system features, learn system defenses, and adapt to successfully execute the attack while evading detection. Thus an autonomous system's cognitive autonomy and cybersecurity intelligence depend on its capability to learn, classify applications (good and bad), predict the attacker's next steps, and remain operational to carryout the mission-critical tasks even under cyberattacks. In this dissertation, we propose novel learning and prediction models for enhancing cognitive autonomy and cybersecurity in autonomous systems. We develop (1) a model using deep learning along with a model selection framework that can classify benign and malicious operating contexts of a system based on performance counters, (2) a deep learning based natural language processing model that uses instruction sequences extracted from the memory to learn and profile the behavior of evasive malware, (3) a scalable deep learning based object detection model with data pre-processing assisted by fuzzy-based clustering, (4) fundamental guiding principles for cognitive autonomy using Artificial Intelligence (AI), (5) a model for privacy-preserving autonomous data analytics, and finally (6) a model for backup and replication based on combinatorial balanced incomplete block design in order to provide continuous availability in mission-critical systems. This research provides effective and computationally efficient deep learning based solutions for detecting evasive cyberattacks and increasing autonomy of a system from application-level to hardware-level. <br>