Reduced Order Modeling (ROM) provides a solid framework for many-query and real-time discretization of physical systems described through parametric Partial Differential Equations (pPDEs). Once few snapshots from the solution manifold are obtained by computationally expensive high-fidelity techniques, such as Finite Element Method or Finite Volume Method, ROM algorithms provide an efficient and reliable methodology to obtain new solutions belonging to some pre-computed low-dimensional space. In this talk, we will focus on advection-dominated and turbulent flows. For such flows, standard linear ROM methods are particularly ill-suited due to the slow-decay of the Kolmogorov $n$-width. Firstly, we will analyze the case of turbulent flows described by Reynolds-averaged Navier-Stokes (RANS) equations that circumvent the difficulties of discretizing all the space-and-time scales of turbulent flows by applying an averaging operator to the Navier-Stokes equations. Nevertheless, RANS equations need to be closed. Historically, such closures where heuristically-derived as additional PDEs. However, recent promising data-driven models have been proposed in this setting. In this talk, we will discuss how to properly define ROM methods for RANS equations closed by data-driven models. Successively, we will investigate how non-linear ROM models are beneficial for advection-dominated flows. In this setting, Scientific Machine Learning (SciML) provides a wide spectrum of available tools, as feed-forward Neural Networks, Graph Neural Networks, etc., to overcome standard issues of classic ROM closures.