GENerative and connected intelligence for 6G Open ManagemEnt (GENOME) research and training programme aims at investigating, developing/fine-tuning and validating large language models (LLMs) for the autonomous management and orchestration of various network technological domains (O-RAN, edge, cloud) through intent-based networking and task-agnostic network functions. It also targets the establishment of a connected intelligence framework for the coordination between these functions via on-the-fly protocol learning to mitigate conflicting management decisions and maximize utilization. Network configuration will also be generated automatically through innovative neuro-symbolic AI algorithms. Moreover, the transparency and resilience of the AI models is researched. All the developed components will be validated via both simulation and testbed experiments, and integrated into a final proof-of-concept. Besides, a detailed training plan is designed including schools, industrial days and soft-skills courses. It involves industrial partners to ensure that career perspectives of ESRs will be significantly increased by participation in the project. High quality dissemination and exploitation activities are also considered.