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Modern society is increasingly surrounded by, and is growing accustomed to, a wide range of Cyber-Physical Systems (CPS), Internet-of-Things (IoT), and smart devices. They often perform safety-critical functions, e.g., personal medical devices, automotive CPS as well as industrial and residential automation, e.g., sensor-alarm combinations. On the lower end of the scale, these devices are small, cheap and specialized sensors and/or actuators. They tend to host small anemic CPUs, have small amounts of memory and run simple software. If such devices are left unprotected, consequences of forged sensor readings or ignored actuation commands can be catastrophic, particularly, in safety-critical settings. This prompts the following three questions: (1) How to trust data produced, or verify that commands were performed, by a simple remote embedded device?, (2) How to bind these actions/results to the execution of expected software? and, (3) Can (1) and (2) be attained even if all software on a device can be modified and/or compromised? In this paper we answer these questions by designing, demonstrating security of, and formally verifying, APEX: an Architecture for Provable Execution. To the best of our knowledge, this is the first of its kind result for low-end embedded systems. Our work has a range of applications, especially, authenticated sensing and trustworthy actuation, which are increasingly relevant in the context of safety-critical systems. APEX is publicly available and our evaluation shows that it incurs low overhead, affordable even for very low-end embedded devices, e.g., those based on TI MSP430 or AVR ATmega processors.