Random fields are not only important in probability, stochastic partial differential equations, and statistics, but also very useful as stochastic models in many scientific areas such as biology, engineering, environmental sciences, geophysical sciences, image processing, and physics. These applications, in turn, have raised many interesting and often challenging questions for statisticians and mathematicians.

For random fields indexed by the Euclidean space or spheres, their important characteristics include smoothness/roughness, self-similarity, isotropy/anisotropy,  and short/long range dependence. Various mathematical and statistical methods have been developed for constructing random fields, and for studying their properties and statistical inferences.

In this talk we present some recent results on statistic and geometric properties of Gaussian random fields and solutions of stochastic partial differential equations (SPDEs).