Abstract: Despite significant development over the last decades, a model able to describe the periphery region of magnetic confinement fusion devices, extending from the edge to the far scrape-off layer, is still missing. The lack of a proper model has undermined our ability to properly simulate the plasma dynamics in this region, which is necessary to predict the heat flux to the vessel wall of future fusion devices, the L-H transition, and ELM dynamics. These are some of the most important issues on the way to a fusion reactor. In this work, a drift-kinetic and a gyrokinetic model able to describe the plasma dynamics in the tokamak periphery are developed, which take into account electrostatic fluctuations at all relevant scales, allowing for comparable amplitudes of background and fluctuating components. In addition, the models implement a full Coulomb collision operator, and are therefore valid at arbitrary collisionality regimes. For an efficient numerical implementation of the models, the resulting kinetic equations are projected onto a Hermite-Laguerre velocity-space polynomial basis, obtaining a moment-hierarchy. The use of systematic closures to truncate the moment-hierarchy equation, such as the semi-collisional closure, allows for the straightforward adjustment of the kinetic physics content of the model. In the electrostatic high collisionality regime, our models are therefore reduced to an improved set of drift-reduced Braginskii equations, which are widely used in scrape-off layer simulations. The first numerical studies based on our models are carried out, shedding light on the interplay between collisional, using the Coulomb collision operator, and collisionless mechanisms. In particular, the dynamics of electron-plasma waves and the drift-wave instability are studied at arbitrary collisionality. A comparison is made with the collisionless limit and simplified collision operators used in state-of-the-art simulation codes, where large deviations in the growth rates and eigenmode spectra are found, especially at the levels of collisionality relevant for present and future magnetic confinement fusion devices.