Let $X$ be a Lévy process and $V$ the reflection at boundaries $0$ and $b>0$. A number of properties of $V$ are studied, with particular emphasis on the behaviour at the upper boundary $b$. The process $V$ can be represented as solution of a Skorokhod problem $V(t)=$ $V(0)+X(t)+L(t)-U(t)$ where $L,U$ are the local times (regulators) at the lower and upper barrier. Explicit forms of $V$ in terms of $X$ are surveyed as well more pragmatic approaches to the construction of $V$, and the stationary distribution $\pi$ is characterised in terms of a two-barrier first passage problem. A key quantity in applications is the loss rate $\ell^b$ at $b$, defined as $\operatorname{Exp}_\pi U(1)$. Various forms of $\ell^b$ and various derivations are presented, and the asymptotics as $b\to\infty$ is exhibited in both the light-tailed and the heavy-tailed regime. The drift zero case $\operatorname{Exp} X(1)=0$ plays a particular role, with Brownian or stable functional limits being a key tool. Further topics include studies of the first hitting time of $b$, central limit theorems and large deviations results for $U$, and a number of explicit calculations for Léevy processes where the jump part is compound Poisson with phase-type jumps.

Keywords: Applied probability, Central limit theorem, Finite buffer problem, First passage problem, Functional limit theorem, Heavy tails, Integro-differential equation, Itô's formula, Linear equations, Local time, Loss rate, Martingale, Overflow, Phase-type distribution, Poisson's equation, Queueing theory, Siegmund duality, Skorokhod problem, Storage process