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One of the major difficulties facing researchers using neural networks is the selection of the proper size and topology of the networks. The problem is even more complex because often when the neural network is trained to very small errors, it may not respond properly for patterns not used in the training process. A partial solution proposed to this problem is to use the least possible number of neurons along with a large number of training patterns. The discussion consists of three main parts: first, different learning algorithms, including the Error Back Propagation (EBP) algorithm, the Levenberg Marquardt (LM) algorithm, and the recently developed Neuron-by-Neuron (NBN) algorithm, are discussed and compared based on several benchmark problems; second, the efficiency of different network topologies, including traditional Multilayer Perceptron (MLP) networks, Bridged Multilayer Perceptron (BMLP) networks, and Fully Connected Cascade (FCC) networks, are evaluated by both theoretical analysis and experimental results; third, the generalization issue is discussed to illustrate the importance of choosing the proper size of neural networks.