Abstract: One of properties of turbulence lies in its huge transport power of mass,  momentum and energy, and in  their large and violent fluctuations of these quantities.  A scalar which is convected by turbulence without reaction to the fluid  is passive scalar and has long been studied in many ways partly because the  transport equation is linear while sharing the convection and molecular diffusion with  those of the velocity.  However, the scalar equation is statistically nonlinear.   It is now well known that the passive scalar fluctuations at small scales are  more intermittent than the velocity and the statistics may not be universal.   In the lecture  the spectrum of the scalar variance at high Schmidt number   is argued from the view points of the spectral theory and the velocity intermittency.  Also the scaling exponents of the moments of the scalar increment   with Schmidt number of order unity are discussed   by using the direct numerical simulation.