When two or more bodies collide, contact occurs between the surfaces of the bodies, preventing them from overlapping in space, which is the primary topic of Chapter 5. Metal formation, vehicle crashes, projectile penetration, various seal designs, and bushing and gear systems are just a few examples of contact phenomena. This chapter is organized as follows. In Sect. 5.2, simple one-point contact examples are presented to illustrate the characteristics of contact phenomena and possible solution strategies. In Sect. 5.3, a general formulation of contact is presented based on the variational formulation. Mathematically, contact problems can be formulated using variational inequalities, which can be converted to constrained optimization problems. These constrained optimization problems are transformed into unconstrained optimization problems using either the penalty method or the Lagrange multiplier method. This transformation allows us to handle the contact constraint separated from constitutive models. Section 5.4 focuses on finite element discretization and numerical integration of the contact variational form. Three-dimensional contact formulation is presented in Sect. 5.5. Due to the highly nonlinear and discontinuous nature of contact problems, considerable care and trial-and-error are required to obtain solutions to practical problems. Section 5.6 presents modeling issues related to contact analysis, such as selecting contact and target bodies, as well as removing rigid-body motions. The second edition significantly extends this chapter, and the two-dimensional and three-dimensional contact algorithms are integrated into NLFEA.