A review is given on studies of high Reynolds number (Re) turbulence by high resolution direct numerical simulations (DNSs) of incompressible homogeneous turbulence with the number of the grid points up to 4096^3.

An emphasis is given on the possible universality of small scale statistics and the Re-dependence of the statistics including

(i) one points statistics of velocity gradients,

(ii) two-point statistics of velocity fields, and

(iii) coarse grained statistics such as energy transfer to subgrid scale, and coarse grained kinetic energy dissipation rate.

Small scale statistics in turbulent shear flows, stratified turbulence, and MHD turbulence are also discussed.