In many speech processing applications, the spectral amplitude is the dominant information while the use of phase spectrum is not so widely spread. In this paper, we present an overview on why speech phase spectrum has been neglected in the conventional techniques used in different applications including: speech separation/enhancement, automatic speech and speaker recognition and speech synthesis. We proceed with giving highlights on the recent progress carried out in demonstrating the importance of phase in different applications and how it impacts on the overall performance. The paper is an introduction to the Interspeech 2014 special session phase importance in speech processing applications.