The joint project FAIR Data Spaces aims to build a common Data Space for industry and science. In doing so, it brings together the two initiatives Gaia-X and the National Research Data Infrastructure (NFDI). But how can these existing initiatives from both worlds be brought together to share data and knowledge across the boundaries of these different domains? How can synergies and fruitful collaborations emerge? FAIR Data Spaces aims to build a shared community by linking the initiatives organizationally, legally, technically, and practically. This enables the creation of building blocks for overarching data spaces based on the common FAIR principles, which will serve as nuclei for a fair data economy. Building on existing infrastructure and data services, the project develops use cases, also called demonstrators, for three data spaces: Biodiversity, Research Data Quality Assurance, and Cross-Platform Data Analysis. The data flows in the individual demonstrators are analysed in terms of data protection and intellectual property rights. The poster presents FAIR Data Spaces and the current project results with focus on the three demonstrators that are discussed in more detail in the following.