This paper presents and discusses algorithms, hardware, and software architecture developed by the TEAM CoSTAR (Collaborative SubTerranean Autonomous Robots), competing in the DARPA Subterranean Challenge. Specifically, it presents the techniques utilized within the Tunnel (2019) and Urban (2020) competitions, where CoSTAR achieved 2nd and 1st place, respectively. We also discuss CoSTAR's demonstrations in Martian-analog surface and subsurface (lava tubes) exploration. The paper introduces our autonomy solution, referred to as NeBula (Networked Belief-aware Perceptual Autonomy). NeBula is an uncertainty-aware framework that aims at enabling resilient and modular autonomy solutions by performing reasoning and decision making in the belief space (space of probability distributions over the robot and world states). We discuss various components of the NeBula framework, including: (i) geometric and semantic environment mapping; (ii) a multi-modal positioning system; (iii) traversability analysis and local planning; (iv) global motion planning and exploration behavior; (i) risk-aware mission planning; (vi) networking and decentralized reasoning; and (vii) learning-enabled adaptation. We discuss the performance of NeBula on several robot types (e.g. wheeled, legged, flying), in various environments. We discuss the specific results and lessons learned from fielding this solution in the challenging courses of the DARPA Subterranean Challenge competition.