Neural network applications in microwave engineering have been reported since the 1990s. Description of artificial neural networks and their key issues, namely architectures, paradigms, training methods, data sets formation, learning and generalization errors, learning speed, etc., in the context of microwave CAD, has been extensively reported. It is clear that neural networks have been widely used for modeling microwave devices and circuits, in many innovative ways. In contrast, the use of neural networks for microwave design by optimization is at a less developed stage. This presentation aims at reviewing the most relevant work in electromagnetics-based design and optimization of microwave circuits exploiting artificial neural networks (ANNs). Measurement-based design of microwave circuits using ANNs is also considered. The conventional and most popular microwave neural optimization approach is reviewed. Advantages and drawbacks of this strategy are emphasized. Improvements of this “black-box” approach such as segmentation, decomposition, hierarchy, design of experiments (DoE) and clusterization are mentioned. The main limitations of the conventional neural optimization approach can be alleviated by incorporating available knowledge into the neural network training scheme. Several innovative strategies are reviewed, including the Difference Method (also called Hybrid EM-ANN), the Prior Knowledge Input (PKI) Method, the Knowledge-Based ANN approach (KBNN), the Neural Space Mapping (NSM) optimization method, the Extended Neural Space Mapping approach, and the Neural Inverse Space Mapping (NISM) optimization algorithm. Practical examples using these techniques are illustrated, including EM-based statistical design of relevant microwave problems. Another strategy for ANN-based design of microwave circuits consists of using synthesis neural networks, also called “inverse neural models”. A synthesis neural network is trained to learn the mapping from the responses to the design parameters of the microwave circuit. Difficulties in developing synthesis neural networks are indicated. Finally, the key issues on transient EM-based design using neural networks are described. Suitable paradigms for approximating nonlinear dynamic behaviors are mentioned, such us Recurrent Neural Networks (RNN) and their corresponding training techniques.ITESO, A.C.