In a Hall Thruster, the [Formula: see text] current generates a strong electron drift with respect to the ions, which can trigger plasma instabilities, such as the electron drift instability (EDI), also called the electron cyclotron drift instability. The EDI has drawn a great deal of attention in the [Formula: see text] community as previous one- and two-dimensional Particle-In-Cell (PIC) studies suggest it plays a major role in the anomalous transport of electrons across the magnetic barrier. However, experiments showed that the EDI has an inherent three-dimensional nature, which cannot be accurately described in a 1D or 2D configuration. Unfortunately, due to their prohibitive computational cost, needed 3D PIC simulations remained inaccessible without scaling plasma parameters, which inevitably modified the physics in a Hall Thruster. Thanks to recent computational developments, this paper presents a 3D fully kinetic investigation of the EDI in a typical Hall Thruster channel and near plume region. The study was conducted using an unstructured grid, thus demonstrating the feasibility to model more complicated geometries in the future. The growth and development of the EDI are described along with other plasma fluctuations, possibly the signature of another plasma instability, the modified-two-stream-instability. 3D effects on the anomalous transport are also assessed and found to be lower than in an analogous 2D simulation. This is due to lower losses in 2D, which lead to the saturation of the EDI occurring at higher energy levels.