Nuclear magnetic resonance spectroscopy (NMR) is heavily employed by chemists and biochemists to study the structures and properties of chemical compounds. The measured data however often contain mixtures of chemicals, subject to changing background and environmental noise. A mathematical problem is to unmix or decompose the measured data into a set of pure or source spectra without knowing the mixing process, a so called blind source separation problem. In the talk, the speaker shall present algorithms for blind separation of spectral mixtures in noisy conditions. The approach combines geometrical and statistical analysis of the data, the geometric approach is based on the vertexes and facets identification of cone structures of the data, while the statistical approach is on decomposing fitting errors when partial knowledge of the source spectra is available. Computational results on data from NMR, Raman spectroscopy and differential optical absorption spectroscopy show the applicability of the methods. This is joint work with Jack Xin from UC Irvine.