The dynamical systems perturbed by weak noise have profound transition behaviors at long time scale. Such rare events has exponentially small probability but critically contribute to lots of interesting physical phenomena, such as phase transition, subcritical instability, etc. This talk mainly focuses on the transition scenarios in
non-gradient systems where no energy landscape exists. Our approach is based on the framework of the large deviation principle (e.g., the Freidlin-Wentzell theory). The applications include the stochastic Lorentz model, Kuramoto-Sivashinsky PDE, etc. The attack of classic problem of turbulent transition, a subcritical bifurcation, will be discussed as one of the most important applications.