The back propagation(BP) algorithm allows multilayer feedforward neural networks to learn input-output mappings from training samples. It iteratively adjusts the network parameters(weights) to minimize the sum of squared approximation errors using a gradient descent technique. However, the mapping acquired through the BP algorithm may be corrupt when errorneous training data are employed. In this paper two types of robust backpropagation algorithms are discussed both from a theoretical point of view and in the case studies of nonlinear regression function estimation and handwritten Korean character recognition. For future research we suggest Bayesian learning approach to neural networks and compare it with two robust backpropagation algorithms.