Changes in weather extremes such as heat waves, cold spells, and heavy precipitation are responsible for a large part of climate-related damage, yet our understanding of these extreme events is limited. Extreme weather events are identified by either the probability distribution function (PDF) of meteorological variables (e.g., temperature, moisture, or precipitation) or the pattern of large-scale circulations (e.g., jet stream meandering or blocking). Yet the probability distribution of these events often displays non-Gaussian tails that are common to the PDFs of dynamical or chemical tracers subject to the advection-diffusion processes, and the tails of these PDFs are particularly important for our understanding of weather extremes. In this talk, I will show examples about how to directly link large-scale atmospheric dynamics to the PDFs of temperature or precipitation. Using a simple advection-diffusion model, the non-Gaussian statistics in temperature can be understood as a result of nonlinear advection of temperature. The effect of large-scale atmospheric advection and associated weather patterns can be measured by the wave activity in atmospheric circulation. It is shown that changes in wave activity can be understood as transport and mixing across a zonal jet, with increased wave activity in a slower zonal jet or vice versa. I will also discuss the application to hydrological cycle in terms of thermodynamic and dynamic contributions over a full probability distribution of precipitation events.