The contact interaction of rod- or beam like structures with three-dimensional (3D) continua (solids) can be found in a variety of different physical problems ranging from classical engineering problems (e.g., belt drives) to biomedical applications (e.g., stents for aortic aneurysm repair). An efficient and accurate modeling approach for the rods or beams is to employ structural beam theories, which reduce the 3D beam continuum to a one-dimensional (1D) curve embedded in 3D space. Employing these 1D beam models in beam-to-solid contact problems requires the development of contact formulations between the 1D beam structures and the 3D solid continuum, i.e., a mixed-dimensional contact formulation. In the context of the employed finite element method, the considered mixed-dimensional contact problem requires suitable contact schemes between the inevitably non-matching grids of the beam centerline and the solid surface. For the discretization and solution of this challenging problem existing techniques from 2D-2D surface-to-surface contact~[1], 1D-1D beam-to-beam contact~[2] and 1D-3D mixed-dimensional coupling~[3], are enhanced and combined. Among the topics addressed in this talk are the different types of contact discretization schemes (e.g., mortar-type, Gauss-point-to-segment) and constraint enforcement strategies. Furthermore, a number of different qualitative and quantitative numerical examples are presented in order to underline the usability of the proposed mixed-dimensional contact schemes for academic and practical applications. [1] P. Wriggers, "Computational contact mechanics" Springer-Verlag Berlin Heidelberg, 2006. [2] C. Meier, A. Popp, and W. A. Wall, “A finite element approach for the line-to-line contact interaction of thin beams with arbitrary orientation” Computer Methods in Applied Mechanics and Engineering, vol. 308, pp. 377–413, 2016. [3] I. Steinbrecher, M. Mayr, M. J. Grill, J. Kremheller, C. Meier, and A. Popp, “A mortar-type finite element approach for embedding 1D beams into 3D solid volumes” Computational Mechanics, vol. 66, Art. no. 6, 2020.