In cut/immersed finite element methods, a major challenge is the accurate numerical integration of cut elements. We first review the idea of a purely data-driven approach for selecting quadrature points and weights to eliminate the need for characterizing the geometry and topology of each cut element—a requirement that makes high-fidelity cut integration methods computationally expensive. We then focus on the moment fitting approach to cut-element quadrature, for which we discuss a suitable objective function and propose a process for generating valid training data. We then investigate two data-driven quadrature methods: (i) a neural network-based approach and (ii) a tensor-product spline interpolation method. We demonstrate that incorporating reduced smoothness into the objective function is essential for achieving sufficient integration accuracy, ensuring that the quadrature error does not dominate the approximation error of a (potentially higher-order accurate) finite element formulation. We highlight the advantage of the spline interpolation method over the neural network approach concerning the imposition of controlled smoothness reduction at specific locations in the parameter space. We analyze the relative error between training and validation data, evaluate the accuracy and computational cost of cut finite element computations with high-fidelity and data-driven quadrature methods, and demonstrate that a data-driven approach enables real-time updates to quadrature rules in cut elements.