The adoption of Deep Neural Networks (DNNs) in several domains allows for increased effectiveness in applications that deal with massive data-intensive and complex data inputs. When employed in safety-critical scenarios, such as automotive, aerospace, healthcare, and autonomous robotics, assessing the DNNs' reliability and functional safety is crucial to ensure their correct in-field operation, even in the presence of hardware faults. However, the system complexity and the massive amounts of data to be processed by DNNs prevent the effective adoption of traditional strategies for reliability characterization and for identifying the most fault-sensitive structures. Accurate fault assessment strategies usually require unacceptable computational power and large evaluation times. On the other hand, faster strategies commonly lack accuracy in correctly representing system faults. Consequently, it is necessary to develop effective strategies that trade-off between performance and accuracy. This work analyses three reliability assessment strategies for deep neural networks and their underlying hardware, highlighting the main solutions and challenges in terms of evaluation performance and fault characterization accuracy. We overview different solutions to evaluate the hardware accelerators implementing DNNs at three abstraction levels: i) by physically injecting faults on a GPU running DNNs, ii) by performing microarchitectural characterization of GPUs to develop applicationaccurate error models, and iii) by using structure-aware crosslayer error modeling on DNN hardware accelerators. Our experimental results indicate that accurate error representation requires structural features from the targeted hardware.