Foundation Models (FMs), such as LLaMA, BERT, GPT, ViT, and CLIP, have demonstrated remarkable success in a wide range of applications, driven by their ability to leverage vast amounts of data for pre-training. However, optimizing FMs often requires access to sensitive data, raising privacy concerns and limiting their applicability in many domains. In this paper, we propose the Federated Foundation Models (FFMs) paradigm, which combines the benefits of FMs and Federated Learning (FL) to enable privacy-preserving and collaborative learning across multiple end-users. We discuss the potential benefits and challenges of integrating FL into the lifespan of FMs, covering pre-training, fine-tuning, and application. We further outline potential future research avenues in FFM, including FFM pre-training, FFM fine-tuning, and federated prompt tuning, which allow the development of more personalized and context-aware models while ensuring data privacy. Moreover, we explore the possibility of continual/lifelong learning in FFMs, as increased computational power at the edge may unlock the potential for optimizing FMs using newly generated private data close to the data source. The proposed FFM concepts offer a flexible and scalable framework for training large language models in a privacy-preserving manner, setting the stage for subsequent advancements in both FM training and federated learning.