The EVERSE project aims to create a framework for research software and code excellence, collaboratively designed and championed by the research communities across five EOSC Science Clusters and national Research Software Expertise Centres, in pursuit of building a European network of Research Software Quality and setting the foundations of a future Virtual Institute for Research Software Excellence. This framework for research software excellence will incorporate aspects involving community curation, quality assessment, and best practices for research software. This collective knowledge will be captured in the Research Software Quality toolkit (RSQkit), a knowledge base to gather and curate expertise that will contribute to high-quality software and code across different disciplines. By embedding the RSQkit and services into the EOSC Science Clusters, EVERSE will demonstrate improvements in the quality of research software and maximise its reuse, leading to standardised software development practices and sustainable research software. Furthermore, we will drive recognition of software and support career progress for developers, from researchers who code to RSEs, raising their capacity to guarantee software quality. The European network for Research Software Quality aims to cross-fertilise different research domains, act as a lobbying organisation, and raise awareness of software as a key enabler in research, with the overall ambition to accelerate research and innovation through improving the quality of research software and code. EVERSE ultimate ambition is to contribute towards a cultural change where research software is recognized as a first-class citizen of the scientific process and the people that contribute to it are credited for their efforts.