Fairness is a critical concept in ethics and social domains, but it is also a challenging property to engineer in software systems. With the increasing use of machine learning in software systems, researchers have been developing techniques to assess the fairness of software systems automatically. Nonetheless, many of these techniques rely upon pre-established fairness definitions, metrics, and criteria, which may fail to encompass the wide-ranging needs and preferences of users and stakeholders. To overcome this limitation, we propose a novel approach, called MODNESS, that enables users to customize and define their fairness concepts using a dedicated modeling environment. Our approach guides the user through the definition of new fairness concepts also in emerging domains, and the specification and composition of metrics for its evaluation through a dedicated domain-specific language. Ultimately, MODNESS generates the source code to implement fair assessment based on these custom definitions. In addition, we elucidate the process we followed to collect and analyze relevant literature on fairness assessment in software engineering (SE). We compare MODNESS with the selected approaches and evaluate how they support the distinguishing features identified by our study. Our findings reveal that i) most of the current approaches do not support user-defined fairness concepts; ii) our approach can cover additional application domains not addressed by currently available tools, e.g., mitigating bias in recommender systems for software engineering and Arduino software component recommendations; iii) MODNESS demonstrates the capability to overcome the limitations of the only two other model-driven engineering-based approaches for fairness assessment.