Compressive sensing has become an active research area in the field of applied and computational harmonic analysis. It has been widely used in signal acquisition and processing, medical imaging, machine learning and many other fields. The classical compressive sensing theory is developed based on the assumption that the sensing data has infinite precision. However, the sensing data is inevitably quantized in practice. An extreme quantization method is only recording the sign of the sensing data. This kind of problem is called 1-bit compressive sensing. In this project we study 1-bit compressive sensing problems, including the signal reconstruction theory and algorithms. In the theory part of this project, our goal is establishing a new mathematical model for 1-bit compressive sensing and prove it can reconstruct the sparse signal with less sensing data and is robust to noise. Moreover, we anticipate developing fast algorithms for 1-bit compressive sensing signal reconstruction and analyzing the convergence of the proposed algorithms.