By moving network functionality from dedicated hardware to software running on end-hosts, Network Functions Virtualization (NFV) pledges the benefits of cloud computing to packet processing. While most of the NFV frameworks today rely on kernel-bypass approaches, no attention has been given to kernel packet processing, which has always proved hard to evolve and to program. In this article, we present Polycube, a software framework whose main goal is to bring the power of NFV to in-kernel packet processing applications, enabling a level of flexibility and customization that was unthinkable before. Polycube enables the creation of arbitrary and complex network function chains, where each function can include an efficient in-kernel data plane and a flexible user-space control plane with strong characteristics of isolation, persistence, and composability. Polycube network functions, called Cubes, can be dynamically generated and injected into the kernel networking stack, without requiring custom kernels or specific kernel modules, simplifying the debugging and introspection, which are two fundamental properties in recent cloud environments. We validate the framework by showing significant improvements over existing applications, and we prove the generality of the Polycube programming model through the implementation of complex use cases such as a network provider for Kubernetes.