The problem of linear independent component analysis (ICA), which is a form of redundancy reduction, was addressed. Following Comon [7], the ICA problem was formulated as the search for a linear transformation that minimizes the mutual information of the resulting components. This is roughly equivalent to finding directions in which negentropy is maximized and which can also be considered projection pursuit directions [16]. The novel approximations of negentropy introduced in [19] were then used for constructing novel contrast (objective) functions for ICA. This resulted in a generalization of the kurtosis-based approach in [7,9], and also enabled estimation of the independent components one by one. The statistical properties of these contrast functions were analyzed in the framework of the linear mixture model, and it was shown that for suitable choices of the contrast functions, the statistical properties were superior to those of the kurtosis-based approach. Next, a new family of algorithms for optimizing the contrast functions were introduced. This was the family of fixed-point algorithms that are not neural in the sense that they are non-adaptive, but share the other benefits of neural learning rules. The main advantage of the fixed-point algorithms is that their convergence can be shown to be very fast (cubic or at least quadratic). Combining the good statistical properties (e.g. robustness) of the new contrast functions, and the good algorithmic properties of the fixed-point algorithm, a very appealing method for ICA was obtained. Simulations as well as applications on real-life data have validated the novel contrast functions and algorithms introduced. Some extensions of the methods introduced in this paper are presented in [20], in which the problem of noisy data is addressed, and in [22], which deals with the situation where there are more independent components than observed variables.