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Provisioning of quality of service (QoS) guarantees has become an increasingly important and challenging topic in the design of the current and the next-generation Internet. The class of envelope processes (EPs) is one of the key elements for many QoS provisioning mechanisms. An arrival EP A(τ) (or a service curve) bounds the cumulative traffic of a flow (or the cumulative service a flow receives) over any interval of length τ. Such bounds can be deterministic or probabilistic, and can be used for provisioning of deterministic or statistical service guarantees. In this article we provide a survey on arrival EPs and service curves. We provide an overview of various EPs proposed in the literature during the last 15 years and discuss their applications and performance in QoS provisioning. We aim to provide a big picture of the existing work. There is considerable research effort addressing QoS issues in resource-constrained access networks (such as wireless networks) and in the new multiprotocol label switching (MPLS) and peer-to-peer (P2P) networking paradigms. We aim to provide a comprehensive survey of existing work, which can yield useful insights, and help the development of new QoS metrics, mechanisms, and architectures for emerging network environments.