The modern society is experiencing sustained economic growth, fueled by technological innovation, globalization, and market expansion. This growth has brought notable improvements in living standards, job creation, and new opportunities. However, this economic boom is accompanied by an increase in acts of fraud. The complexity of financial transactions, increased accessibility to personal data, and the globalization of markets offer fraudsters new opportunities to exploit vulnerabilities in systems. Credit card fraud, insurance fraud, tax fraud, e-commerce fraud, and cybercrime are among the most common types of fraud. It is in the face of this growing scourge that the implementation of fraud detection systems is essential. These systems, equipped with sophisticated analytical tools and intelligent algorithms, help identify suspicious behavior and prevent fraudulent transactions. They constitute a formidable weapon for fighting financial crime and protecting the interests of economic actors. Over the past twenty years, research into anomaly detection has grown exponentially, driven by the exploration of statistical models, artificial intelligence, and machine learning. Among these models, supervised learning algorithms have long dominated the scene. However, they raise challenges that more recent semi-supervised and unsupervised learning models can overcome. This study takes an in-depth look at the most successful anomaly detection techniques in the field of financial fraud, highlighting recent advances in semi-supervised and unsupervised learning.