Blind Source Separation (BSS) is a statistical signal processing technique and has recently been developed for many applications. The aim of this thesis is to investigate the blind signal separation problem under the environment where noise, reverberation and nonlinear distortion exist in the mixture and to develop novel solutions to solve the problem. The success and efficacy of the proposed algorithms is analysed in terms of robustness to noise, accuracy of recovered signal and speed of convergence. Linear BSS algorithms for instantaneous and convolutive mixtures are investigated and tested by a set of specially designed simulated experiments under various conditions. In addition, the post-nonlinear instantaneous mixture model has been critically researched and the theory of signal separability has been established. To overcome the limitation and drawbacks of the existing works on post-nonlinear mixture, a novel solution has been developed to separate noisy post-nonlinear instantaneous mixtures of non-stationary and temporally correlated sources and this work further extends to the case of noisy convolutive mixture. The proposed models allow source non-stationarity and temporal correlation to be incorporated into the new solutions. The Maximum Likelihood (ML) approach has been developed for both of the proposed algorithms to estimate the model parameters by the Expectation Maximisation (EM) algorithm and the post-nonlinearity is estimated by a set of self-updating polynomials whose coefficients are updated as part of the model parameters. The theoretical foundation of the proposed solutions has been rigorously developed and discussed in detail. The new algorithms have been tested by simulations using both synthetically generated and recorded speech signals to verify the accuracy and efficacy. The results show that the proposed algorithms outperform existing algorithms in the separation performance where significant improvement has been obtained.