Interacting particle systems provide flexible and powerful models that are useful in many application areas such as sociology (agents), molecular dynamics (proteins), etc. However, particle systems with large numbers of particles are very complex and difficult to handle, both analytically and computationally. Therefore, a common strategy is to derive effective equations that describe the time evolution of the empirical particle density. Our aim is to derive and study continuum models for the mesoscopic behavior of particle systems. In particular, we are interested in finite size effects. We will introduce nonlinear and non-Gaussian models that provide a more faithful representation of the evolution of the empirical density of a given particle system than the usual linear Gaussian perturbations around the hydrodynamic limit models. We want to study the well-posedness of these nonlinear SPDE models and to control the weak error of the SPDE approximation. A prototypical example that we will consider is the formal identification of a finite system of diffusions with the singular Dean-Kawasaki SPDE. This is the joint work with H. Kremp and N. Perkowski. Furthermore, we will discuss the application of these types of equations in the feedback-loop opinion dynamics. This is a joint work with N. Dj. Conrad and Jonas Köppl.