Deep Learning-based Reduced Order Models (DL-ROMs) have recently emerged in ROM literature as they are able to extremely decrease the computational complexity related to the solution of a large class of differential problems that comprise, e.g., nonlinearities, time dependency and non-affine terms. The success of DL-ROMs is also due to them inheriting the characteristic features of their neural network core, namely, outstanding approximation capabilities, remarkable accuracy, and extremely fast simulation times. However, despite their promising results with problems characterized by physical variability, DL-ROMs have never been applied to problems encompassing parametric domains. In this context, we aim at widening the range of application of DL-ROMs to problems featuring geometrical variability. More precisely, our purpose is twofold: firstly, we focus on the development and the analysis of two different paradigms (respectively based on adaptive and ad hoc basis functions) to inform the neural network architecture with a priori geometrical knowledge. Secondly, we stress the importance of developing a mesh-free continuous counterpart of classical DL-ROMs defined in a finite-dimensional setting to deal with generic datasets comprising, e.g., multi-resolution or missing data, which are common features in geometrically parameterized problems. Ultimately, this talk will focus on the introduction of Continuous Geometry-Aware DL- ROMs (CGA-DL-ROMs), a novel family of techniques that combines the expressive power of geometry-aware paradigms with the flexibility of mesh-free architectures in a DL-ROM context, paving the way to the application of deep learning-based ROMs to a larger class of problems possibly arising from industrial challenges and real-world applications.