How to protect people from algorithmic harms? A promising solution, although in its infancy, is algorithmic impact assessment (AIA). AIAs are iterative processes used to investigate the possible short and long-term societal impacts of AI systems before their use, but with ongoing monitoring and periodic revisiting even after their implementation. When conducted in a participatory and trans-parent fashion, they could create bridges across the legal, social and computer science domains, promoting the accountability of the entity performing them as well as public scrutiny. They could enable to re-attach the societal and regulatory context to the mathematical- cal definition of fairness, thus expanding the formalistic approach thereto. Whilst the regulatory framework in the European Union currently lacks the obligation to perform such AIA, some other provisions are expected to play a role in AI development, leading the way towards more widespread adoption of AIA. These include the Data Protection Impact Assessment (DPIA) under the General Data Protection Regulation (GDPR), the risk assessment process under the Digital Services Act (DSA) and the Conformity Assessment (CA) foreseen under the AI Regulation proposal. In this paper, after briefly introducing the plurality of definitions of fairness in the legal, social and computer science domains, and ex- explaining to which extent the current and upcoming legal framework mandates the adoption of fairness metrics, we will illustrate how AIA could create bridges between all these disciplines, allowing us to build fairer AI solutions. We will then recognise the role of DPIA, DSA risk assessment and CA by discussing the contributions they can offer towards AIA but also identify the aspects lacking therein. We will then identify how these assessment provisions could aid the overall technical discussion of introducing and assessing fairness in AI-based models and processes