Autonomous, agile quadrotor flight raises fundamental challenges for robotics research in terms of perception, planning, learning, and control. A versatile and standardized platform is needed to accelerate research and let practitioners focus on the core problems. To this end, we present Agilicious, a co-designed hardware and software framework tailored to autonomous, agile quadrotor flight. It is completely open-source and open-hardware and supports both model-based and neural-network--based controllers. Also, it provides high thrust-to-weight and torque-to-inertia ratios for agility, onboard vision sensors, GPU-accelerated compute hardware for real-time perception and neural-network inference, a real-time flight controller, and a versatile software stack. In contrast to existing frameworks, Agilicious offers a unique combination of flexible software stack and high-performance hardware. We compare Agilicious with prior works and demonstrate it on different agile tasks, using both model-based and neural-network--based controllers. Our demonstrators include trajectory tracking at up to 5g and 70 km/h in a motion-capture system, and vision-based acrobatic flight and obstacle avoidance in both structured and unstructured environments using solely onboard perception. Finally, we demonstrate its use for hardware-in-the-loop simulation in virtual-reality environments. Thanks to its versatility, we believe that Agilicious supports the next generation of scientific and industrial quadrotor research.