Physical Unclonable Functions (PUF) are of increasing importance due to their many hardware security applications including chip fingerprinting, metering, authentication, anticounterfeiting, and supply-chain tracing, e.g., DARPA SHIELD. This paper presents BIST-PUF, the first built-in-self-test (BIST) methodology for online evaluation of weak and strong PUFs. BIST-PUF provides a paradigm shift in the evaluation of the unclonable circuit identifiers: unlike earlier known PUF evaluation suites that are software-based and offline, BIST-PUF enables onthe-fly assessment of the desired PUF properties all in hardware. More specifically, the BIST-PUF structure is designed to evaluate two main properties of PUFs, namely unpredictability and stability. These properties are important for ensuring robustness and security in face of operational, structural, and environmental fluctuations due to variations, aging or adversarial acts. For BIST-PUF unpredictability evaluation, we identify and adopt the tests of randomness that are amenable to hardware implementation. For stability assessment, the BIST-PUF suggests three distinct methods, namely, sensor-based, parametric interrogation, and multiple interrogations. Proof-of-concept implementation of the BIST-PUF in FPGA demonstrates its low overhead, effectiveness, and practicality.