Chemotaxis is characterized by the directional cell movement following external chemical gradients. It plays a crucial role in a variety of biological processes including neuronal development, wound healing and cancer metastasis. Ultimately, the accuracy of gradient sensing is limited by the fluctuations of signaling components, e.g. the stochastic receptor occupancy on cell surface. We use concepts and techniques from interrelated disciplines (statistics, information theory, and statistical physics) to model the stochastic information processing in eukaryotic chemotaxis. Specifically, we address the following issues: \begin{enumerate} \item What are the physical limits of the gradient estimation? \\ \item How much information can be reliably gained by a chemotaxing cell? \\ \item How to optimize the chemotactic performance? \\ \end{enumerate} Through answering those questions, we expect to derive extra insights for general biological signaling systems.