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Currently, Machine Learning (ML) is becoming ubiquitous in everyday life. Deep Learning (DL) is already present in many applications ranging from computer vision for medicine to autonomous driving of modern cars as well as other sectors in security, healthcare, and finance. However, to achieve impressive performance, these algorithms employ very deep networks, requiring a significant computational power, both during the training and inference time. A single inference of a DL model may require billions of multiply-and-accumulated operations, making the DL extremely compute- and energy-hungry. In a scenario where several sophisticated algorithms need to be executed with limited energy and low latency, the need for cost-effective hardware platforms capable of implementing energy-efficient DL execution arises. This paper first introduces the key properties of two brain-inspired models like Deep Neural Network (DNN), and Spiking Neural Network (SNN), and then analyzes techniques to produce efficient and high-performance designs. This work summarizes and compares the works for four leading platforms for the execution of algorithms such as CPU, GPU, FPGA and ASIC describing the main solutions of the state-of-the-art, giving much prominence to the last two solutions since they offer greater design flexibility and bear the potential of high energy-efficiency, especially for the inference process. In addition to hardware solutions, this paper discusses some of the important security issues that these DNN and SNN models may have during their execution, and offers a comprehensive section on benchmarking, explaining how to assess the quality of different networks and hardware systems designed for them.