Two-degree-of-freedom (2DOF) systems play an important role in many areas of mechanical engineering. There are two groups of structural compositions of 2DOF systems. One of these groups consists of two rigid bodies (masses) connected to each other by one or two (in parallel) links (springs and/or dashpots). In this paper this group is considered as the basic 2DOF system. The other structural composition of a 2DOF system consists of a basic system, in which one or both masses are connected to non-movable supports. It should be stressed that these connecting links allow relative motion between the two masses. Each mass in a 2DOF system moves independently according to its law of motion. Therefore, in 2DOF systems the values of the forces that the connecting links exert toward the masses are proportional to the di erences between the absolute values of the displacements and/or velocities of the masses. This is contrary to 1DOF systems, where these links exert forces proportional to the absolute values of the corresponding parameters of motion. A survey of published sources from the last seventy- ve years shows that all published pairs of di erential equations attempting to describe the motion of 2DOF systems actually contain a mix of mathematical terms representing the absolute values of the forces exerted by the connecting links as well as the di erences of these forces. None of the surveyed sources contained any rigorous solutions of any pair of simultaneous di erential equations of motion of 2DOF systems. Accounting for the relative motion between the masses of 2DOF systems, I revised the di erential equations from the surveyed publications and also composed the additional pairs of di erential equations for all practically relevant 2DOF systems and the possible varieties of their structures. Applying the Laplace Transform methodology to these di erential equations of motion, I obtained rigorous mathematical solutions for all of them. Several examples are presented in this paper. Introduction Movable mechanical structures predominantly consist of one- and two-degree-of-freedom (1DOF and 2DOF) systems. The investigation of the operational processes of these systems is based on the analysis of their basic parameters of motion. These parameters can be determined by composing and solving the appropriate di erential equations of motion. My book [1] addresses the entire spectrum of the solutions of second-order linear di erential equations of motion of 1DOF mechanical systems. This book contains the compositions and solutions of all possible linear second- order di erential equations of motion for common systems. The diversity of these versions is based on utilizing all possible combinations of mathematical terms that represent loading factors. The solutions of these di erential equations were obtained by using the Laplace Transform methodology. 1 Currently, I am completing a similar work for 2DOF mechanical systems. Hence, I carried out a survey of published sources in the eld of composing and solving second-order linear di erential equations of motion of 2DOF systems, paying special attention to the structures of mathematical terms that depend on displacements and velocities of the masses of the system. This survey includes publications that were issued during the last seventy- ve years. All 2DOF systems consist of two masses connected to each other by links (elastic links such as springs and uid links such as dashpots). This paper classi es 2DOF systems by the number of links they have. In a one-link system, the masses are only connected to each other. In surveyed publications, this is sometimes referred to as an `unrestricted', `unconstrained', or `ungrounded' system. In a two-link system, the masses are connected to each other, and one of the masses is also connected to a non-movable support. In surveyed publications, this is sometimes referred to as a `restricted', `constrained', or `grounded' system. Finally, in a three-link system, both masses are connected to unmovable supports, as well as to each other. In surveyed publications, this also falls in the category of `restricted' systems. It should be noted that a link can consist of a spring, a dashpot, or both in parallel. The links between the masses allow each mass to move independently according to its particular law of motion. Therefore, in a 2DOF system the two masses constantly perform relative motion between each other. The only alternative way of motion for these two masses is to perform simultaneous motion due to having identical laws of motion. In this case, these two masses move as one body, and, therefore, the two masses constitute a 1DOF system. This allows us to emphasize that the concept of relative motion represents one of the fundamental characteristics of the kinematics of motion of the masses of any 2DOF system. Each mass of a 2DOF system can be independently assigned any initial conditions of motion and each mass can be subjected to any external forces. The motion of a 2DOF system can be described by a pair of simultaneous di erential equations (one equation for each of the two masses). By expanding the Laplace Transform methodology with a few conventional algebraic procedures I solved the vast majority of published pairs of simultane- ous di erential equations of motion of unrestricted (one-link) 2DOF systems. A few examples of solutions for these systems are presented in my other publication [2]. However, the aforementioned survey made it clear that the overwhelming majority of publications deal with restricted systems. Therefore, the survey of publications was focused on clarifying the basics of the kinematics of mo- tion of the masses in restricted systems, and on the structures of the mathematical terms that constitute the second order linear di erential equations of motion of these systems as well as on the methodologies of solving them. The survey shows that just a few related publications belong to the area of mathematics as, for instance, [3] - [4]. These publications, like all other surveyed publications, present the same pairs of di erential equations without any attempts of getting their solutions. It should be noted that the overwhelming majority of surveyed publications belong to the theory of vibrations [5] - [16]. Therefore, the list of