This paper proposes a novel training mechanism called collaborative edge training for big AI models in wireless networks. The goal is to address the challenges of training large AI models on edge devices, such as limited computational resources and privacy concerns. Traditional methods rely on centralized cloud training or on-device training, but these approaches have limitations in terms of sustainability and privacy. Collaborative edge training leverages a pool of trusted edge devices to accelerate and sustain big AI model training at the edge. The paper presents a comprehensive framework for collaborative edge training, analyzing its merits and sustainable scheduling choices. It also investigates the impact of parallelism design on training energy consumption through empirical studies. The framework includes four phases: resource selection, orchestration strategy, model replication, and training execution. The system allows for efficient utilization of available edge resources, enabling more fault tolerance and training robustness compared to single-device training. The paper discusses four key research questions (RQs) related to collaborative edge training: participant selection, parallelism design, device topology arrangement, and fault tolerance. The results show that collaborative training can significantly accelerate model training and achieve energy efficiency comparable to single-device training. Data parallelism and pipeline parallelism are found to be particularly effective in reducing energy consumption and improving training throughput. The paper also discusses open challenges for sustainable collaborative edge training, including sustainability metric design, efficient collaboration orchestration, participant incentivization, AI-native wireless networks, wireless-native AI models, power-efficient hardware, and practical privacy and security. These challenges highlight the need for further research to ensure the sustainability and effectiveness of collaborative edge training in wireless networks.This paper proposes a novel training mechanism called collaborative edge training for big AI models in wireless networks. The goal is to address the challenges of training large AI models on edge devices, such as limited computational resources and privacy concerns. Traditional methods rely on centralized cloud training or on-device training, but these approaches have limitations in terms of sustainability and privacy. Collaborative edge training leverages a pool of trusted edge devices to accelerate and sustain big AI model training at the edge. The paper presents a comprehensive framework for collaborative edge training, analyzing its merits and sustainable scheduling choices. It also investigates the impact of parallelism design on training energy consumption through empirical studies. The framework includes four phases: resource selection, orchestration strategy, model replication, and training execution. The system allows for efficient utilization of available edge resources, enabling more fault tolerance and training robustness compared to single-device training. The paper discusses four key research questions (RQs) related to collaborative edge training: participant selection, parallelism design, device topology arrangement, and fault tolerance. The results show that collaborative training can significantly accelerate model training and achieve energy efficiency comparable to single-device training. Data parallelism and pipeline parallelism are found to be particularly effective in reducing energy consumption and improving training throughput. The paper also discusses open challenges for sustainable collaborative edge training, including sustainability metric design, efficient collaboration orchestration, participant incentivization, AI-native wireless networks, wireless-native AI models, power-efficient hardware, and practical privacy and security. These challenges highlight the need for further research to ensure the sustainability and effectiveness of collaborative edge training in wireless networks.