Biological neural networks are systems of extraordinary computational capabilities shaped by evolution, development, and lifetime learning. The interplay of these elements leads to the emergence of adaptive behavior and intelligence, but the complexity of the whole system of interactions is an obstacle to the understanding of the key factors at play. Inspired by such intricate natural phenomena, Evolved Plastic Artificial Neural Networks (EPANNs) use simulated evolution in-silico to breed plastic neural networks, artificial systems composed of sensors, outputs, and plastic components that change in response to sensory-output experiences in an environment. These systems may reveal key algorithmic ingredients of adaptation, autonomously discover novel adaptive algorithms, and lead to hypotheses on the emergence of biological adaptation. EPANNs have seen considerable progress over the last two decades. Current scientific and technological advances in artificial neural networks are now setting the conditions for radically new approaches and results. In particular, the limitations of hand-designed structures and algorithms currently used in most deep neural networks could be overcome by more flexible and innovative solutions. This paper brings together a variety of inspiring ideas that define the field of EPANNs. The main computational methods and results are reviewed. Finally, new opportunities and developments are presented.