Generative artificial intelligence (AI) offers numerous opportunities for research and innovation, but its commercialization has raised concerns about the transparency and safety of frontier AI models. Most models lack the necessary components for full understanding, auditing, and reproducibility, and some model producers use restrictive licenses whilst claiming that their models are "open source". To address these concerns, we introduce the Model Openness Framework (MOF), a three-tiered ranked classification system that rates machine learning models based on their completeness and openness, following open science principles. For each MOF class, we specify code, data, and documentation components of the model development lifecycle that must be released and under which open licenses. In addition, the Model Openness Tool (MOT) provides a user-friendly reference implementation to evaluate the openness and completeness of models against the MOF classification system. Together, the MOF and MOT provide timely practical guidance for (i) model producers to enhance the openness and completeness of their publicly-released models, and (ii) model consumers to identify open models and their constituent components that can be permissively used, studied, modified, and redistributed. Through the MOF, we seek to establish completeness and openness as core tenets of responsible AI research and development, and to promote best practices in the burgeoning open AI ecosystem.