Simulating fluid flows in different virtual scenarios is of key importance in engineering applications. However, high-fidelity, full-order models relying, e.g., on the finite element method, are unaffordable whenever fluid flows must be simulated in almost real-time. Reduced order models (ROMs) relying, e.g., on proper orthogonal decomposition (POD) provide reliable approximations to parameter-dependent fluid dynamics problems in rapid times. However, they might require expensive hyper-reduction strategies for handling parameterized nonlinear terms, and enriched reduced spaces (or Petrov–Galerkin projections) if a mixed velocity–pressure formulation is considered, possibly hampering the evaluation of reliable solutions in real-time. Dealing with fluid–structure interactions entails even greater difficulties. The proposed deep learning (DL)-based ROMs overcome all these limitations by learning, in a nonintrusive way, both the nonlinear trial manifold and the reduced dynamics. To do so, they rely on deep neural networks, after performing a former dimensionality reduction through POD, enhancing their training times substantially. The resulting POD-DL-ROMs are shown to provide accurate results in almost real-time for the flow around a cylinder benchmark, the fluid–structure interaction between an elastic beam attached to a fixed, rigid block and a laminar incompressible flow, and the blood flow in a cerebral aneurysm.