This book focuses on the finite element method in fluid flows. It is targeted at researchers, from those just starting out up to practitioners with some experience. Part I is devoted to the beginners who are already familiar with elementary calculus. Precise concepts of the finite element method remitted in the field of analysis of fluid flow are stated, starting with spring structures which are most suitable to show the concepts of superposition/assembling. Pipeline system and potential flow sections show the linear problem. Advection diffusion section presents time dependent problem. The mixed interpolation is explained using creeping flows. The elementary computer programs by FORTRAN is included. Part II provides information of the recent computational methods and their applications to the practical problems. Theories of SUPG (Streamline-Upwind/Petrov-Galerkin) formulation, characteristic formulation and ALE (Arbitrary Lagrangian-Eulerian) formulation and others are presented with practical results solved by those methods. For reasons who are not familiar with indicial notation, those descriptions are attached in the appendix.