The rapid advancement of artificial intelligence (AI) technology is transforming various sectors, including finance. The EU AI Act introduces specific regulations governing the development and use of AI systems. This paper explores the implications of the AI Act for financial institutions, focusing on how different AI systems are categorized by risk levels and the responsibilities of financial institutions in their development or usage. The authors analyse real use cases of AI in the financial sector, such as credit scoring systems and chatbots, through the lens of the AI Act requirements. Finally, the study offers practical recommendations to help financial institutions align with the new regulatory landscape.