The Conforming Decomposition Finite Element Methods (CDFEM)[1] and Conforming Transient h-r Unstructured Adaptive Mesh Refinement (cThruAMR) methods are discretization methods for coupled multi-phase and multi-material problems with moving interfaces. In these methods, a combination of h-adaptivity and r-adaptivity are employed to automatically generate good quality, interface-conforming meshes. Weak and strong discontinuities are captured by using single-valued or double-valued fields, respectively, on the nodes on the interface. Because the resulting elements conform to the interfaces, standard finite element formulations can be employed, and optimal convergence rates are guaranteed. Traditionally, the interface evolution in CDFEM and cThruAMR has been described using level set methods. First and second order time-accurate schemes have been developed by coupling the level set evolution to the fluid continuity and momentum equations. Multiple issues have been identified with the use of level set methods, however. Frequently, the interface resolution requirements can exceed that of the continuity and momentum equations. Also, level set fields must be reinitialized to preserve the signed distance property. In our experience, the frequency of reinitialization can significantly impact the accuracy and robustness of the overall method. If reinitialization is too infrequent, the interface location exhibits spurious behaviour. Too frequent reinitialization is also detrimental because it can degrade the time accuracy of the overall scheme. Additionally, level set methods require an advection velocity to be specified over the entire domain. If the local fluid velocity is used for advection, there can be rapid degradation of the signed distance property of the level set field. Consequently, we are pursuing alternative descriptions of interface evolution. In this talk we present our progress toward a coupled adaptive semi-Lagrangian method[2] with CDFEM and cThruAMR for multi-phase and multi-material applications. Adaptive semi-Lagrangian methods provide a means of decoupling the interface and physics mesh resolution, do not require reinitialization, and only require the velocity to be specified on the interface. Multiple benchmark problems are used to demonstrate the quality of the interface-conforming discretizations and the resulting numerical accuracy.