The pseudo-deterministic (or simply deterministic) transport method is used in Monte Carlo particle transport problems to increase the sampling of a region in space that particles have a low probability of reaching. Although it has been used by many authors over the years, mainly due to its implementation in the code MCNP which itself has been extensively validated, to our knowledge, a proof of the unbiasedness of the method has never been published. This article thus provides a comprehensive mathematical description of the pseudo-deterministic transport method, built from simple transformations of a given random variable representing an arbitrary physical quantity of interest. Some considerations are made for the handling of potential secondary particles created during the interaction process, and of specific estimators such as pulse height or energy spectrum. This description should allow for a better understanding of the technique, including its possible uses and limitations, and can be used as a reference by both simulation code users and developers looking to implement pseudo-deterministic transport in their code.