Zero-knowledge proofs enable proving a statement without revealing any information beyond its truth. This paradoxical notion has evolved over the last few decades from a theoretical concept to the wide adoption of highly efficient zero-knowledge proof systems in practice. At the forefront of this development are proof systems called zk-SNARKs, which stands for zero-knowledge succinct non-interactive argument of knowledge. Not only do they avoid multiple rounds of interaction, but zk-SNARKs also offer succinct proofs whose length is much shorter than the size of the proved statement, with some constructions even achieving constant-size proofs. Among the most recent state-of-the-art constructions is the zk-SNARK "PLONK" by Gabizon, Williamson, and Ciobotaru from 2019. It has constant-size proofs of only half a kilobyte and sublinear proof verification time. Furthermore, it only requires a single trusted setup of its public parameters to support proofs of any statement up to a certain size bound, making PLONK a universal and fully succinct zk-SNARK. Although highly influential and implemented in several real-world applications, there is no formal security proof of its zero knowledge property. In this thesis, we disclose a vulnerability found in PLONK's implementation of zero knowledge and propose how to fix it. As a result, the PLONK protocol has been patched accordingly. Our primary contribution is a formal security proof establishing that the resulting version of PLONK achieves statistical zero knowledge. Towards this goal, we show how to simulate proofs up to an exponentially small difference without relying on any secret information used by the prover. Following the standard definition of zero knowledge, this implies that PLONK proofs reveal (statistically) zero information beyond the truth of the statement. Moreover, we conduct a rigorous security analysis of the entire PLONK protocol, proving the security of all its underlying components. This allows us to show a precise upper bound on PLONK's knowledge soundness error in the algebraic group model. Since the original proof given by the authors of PLONK relies on the same idealized model, our results help towards a better understanding of the security guarantees of PLONK in general.