Nowadays, the number of Integrated Circuits (ICs) used in safety- and mission-critical applications, i.e., automotive, smart health-care, defense, critical infrastructure, etc., is ever increasing. These applications demand that ICs carry functional safety properties. In this thesis, we develop a Built-In Self Test (BIST) approach for Analog and Mixed-Signal (A/M-S) ICs, called Symmetry-Based Built-In Self Test (SymBIST), which achieves several objectives towards the functional safety goal. SymBIST is a generic BIST paradigm based on identifying inherent in- variances and constructing them by processing internal signals. By construction, invariances should hold true only in error-free operation, while their violation points to abnormal operation. The invariances are being checked using dedicated on-die checkers. SymBIST is a one-stop solution for three different functional safety goals. First, it can be used for defect-oriented test with high defect coverage towards post-manufacturing test quality improvement. Second, it can be applied for on-line test concurrently with the operation towards detecting aging, latent defects, and single event upsets. Third, it can be used for fault diagnosis with high diagnosis resolution and small diagnosis cycle towards yield learning and implementing corrective actions to avoid failure re-occurrence. We demonstrate SymBIST on an industrial Successive Approximation Register (SAR) Analog-to-Digital Converter (ADC) Intellectual Property (IP). We also demonstrate the design of on-chip digital test stimulus generators and minimal re-configurations required in the defect-oriented and diagnosis operation modes. For the defect-oriented test use case, SymBIST offers a fast time in the order of sub-micro-seconds and Likelihood-Weighted (L-W) defect coverage of over 86%. For the on-line test use case, SymBIST checks the invariances in real-time without interrupting the operation of the circuit and irrespective of the input and detects aging, latent defects, and transient errors. For the diagnosis use case, the SymBIST response is used as a digital diagnostic measurement. To reduce defect ambiguity, we use the same test infrastructure, but with different SymBIST setups. SymBIST shows high diagnosis resolution, i.e., 73% correct diagnosis of defects while over 96% of defects are in ambiguity groups of maximum size 5, and offers a fast diagnosis cycle in the order of a few micro-seconds. Last but not least, SymBIST features no performance penalty, an area overhead of around 5%, and has a fully digital interface making it compatible with modern digital test access mechanisms based on two external pins.