In this talk, we introduce the Navier-Stokes system and the Euler system, which are mathematical models of fluid dynamics. After explaining both models, we consider the vanishing viscosity limit of solutions. Precisely, we may expect that the solution of the Navier-Stokes system converges to that of the Euler system by taking the limit on the viscosity (some parameter). Concerning this question, we give a simple overview of the well-known result obtained by Kato (J. Funct. Anal. 9 (1972), 296--305). Finally, as its related topics, we introduce the speaker's recent result if there is time.