many patterns of cell and tissue organization are specified during development by gradients of morphogens, substances that assign different cell fates at different concentrations. one of the central questions in cell and developmental biology is to identify mechanisms by which the morphogen gradient systems might achieve robustness to ensure reproducible embryonic patterns despite genetic or environmental fluctuations. \vskip .1in \noindent recently, through computations and analysis of various bio-chemical models and examination of old and new experimental data, we found a set of new mechanisms for enhancing robustness of cell-cell signaling through non-signaling cell surface molecules (e.g., hspg). in addition, we examined the roles of diffusive ligands (e.g., sog) on the formation and robustness of bmp (bone morphogenetic protein) gradients in the drosophila embryo. in this talk, i shall also discuss some mathematical and computational challenges associated with such study, and present a new class of numerical algorithms for reaction-diffusion equations arising from biological models.