Recent advances in text-to-video generation have sparked interest in generative video editing tasks. Previous methods often rely on task-specific architectures (e.g., additional adapter modules) or dedicated customizations (e.g., DDIM inversion), which limit the integration of versatile editing conditions and the unification of various editing tasks. In this paper, we introduce UNified In-Context Video Editing (UNIC), a simple yet effective framework that unifies diverse video editing tasks within a single model in an in-context manner. To achieve this unification, we represent the inputs of various video editing tasks as three types of tokens: the source video tokens, the noisy video latent, and the multi-modal conditioning tokens that vary according to the specific editing task. Based on this formulation, our key insight is to integrate these three types into a single consecutive token sequence and jointly model them using the native attention operations of DiT, thereby eliminating the need for task-specific adapter designs. Nevertheless, direct task unification under this framework is challenging, leading to severe token collisions and task confusion due to the varying video lengths and diverse condition modalities across tasks. To address these, we introduce task-aware RoPE to facilitate consistent temporal positional encoding, and condition bias that enables the model to clearly differentiate different editing tasks. This allows our approach to adaptively perform different video editing tasks by referring the source video and varying condition tokens "in context", and support flexible task composition. To validate our method, we construct a unified video editing benchmark containing six representative video editing tasks. Results demonstrate that our unified approach achieves superior performance on each task and exhibits emergent task composition abilities.