This dissertation deals with mathematical modeling of flow and transport in random domains. Such problems arise when boundaries of simulation domains are either uncertain or fluctuate randomly in time or both. Examples of such problems include flow in micro-channels, micro-fin exchangers, and biological systems. In these and other applications, random domain geometry affects flow and/or transport behavior and heat efficiency. We use a numerical algorithm consisting of three steps to solve random domain problems with. The first step is to use a finite-term expansion (e.g., a Karhunen-Loeve or Fourier expansion) to parameterize random surfaces (or roughness) of micro- vessels and micro-channels. The second step is to use stochastic mapping, which transforms a deterministic governing equation in a random domain into a stochastic equation in a deterministic domain. The final step is to use either a stochastic Galerkin method or a stochastic collocation method to represent a solution of the stochastic equation as polynomial chaos expansions in terms of orthogonal polynomials. The polynomial type is dependent upon the random variable representing the uncertain data/parameter. This general procedure is used to investigate the effect of endothelial roughness on blood flow, the impact of temporal and spatial fluctuations of cell free layer on nitric oxide production and scavenging, and Stokes flow in domains bounded by surfaces with roughness under a periodic condition