We are all familiar with the sound which can be viewed as a wave motion in air or other elastic media. In this case, sound is a stimulus. Sound can also be viewed as an excitation of the hearing mechanism that results in the perception of sound. The interaction between the physical properties of sound, and our perception of them, poses delicate and complex issues. It is this complexity in audio and acoustics that creates such interesting problems. Acoustic echo is inevitable whenever a speaker is placed near to a microphone in a general full-duplex communication application. The most common communication scenario is the hands-free mobile communication kits for a car. For example, the voice from the loudspeaker is unavoidably picked up by the microphone and transmitted back to the remote speaker. This makes the remote speaker hear his/her own voice distorted and delayed by the communication channel or called end to end delay, which is known as echo. Obviously, the longer the channel delay, the more annoying the echo resulting a decrease in the perceived quality of the communication service such as VoIP conference call. In the thesis, we propose to use different approaches to perform acoustic echo cancellation. In addition, we exploit the idea of blind source separation (BSS) which can estimate source signals using only information about their mixtures observed in each input signal. In addition, we provide a wide theoretical analysis of models and algorithmic aspects of the widely used adaptive algorithm Least Mean Square (LMS). We compare these with Non-negative Matrix Factorization (NMF), and their various extensions and modifications, especially for the purpose of performing AEC by employing techniques developed for monaural sound source separation.