In many multi-physics applications, coupled models and dimension- reduced PDE systems play an important role in the mathematical modelling of physical effects on different scales, e.g., fractures of co- dimension 1 in porous media systems, networks of co-dimension 2 or point sources. Although these simplified models seem very attractive from the computational point of view, for coupled problems they may result in a solution of reduced regularity. Mathematically, transmission problems with piecewise smooth solutions or PDEs in a distributional sense with singular solutions arise. The global accuracy is often dominated by local effects at the inter- faces, and local singularities can pollute the numerical solution. Here we discuss several issues such as hierarchical limiting techniques, local energy corrections and optimal estimates for the flux variables. The coupling is controlled by pairs of balance equations providing a very flexible framework. But stability and accuracy do not come for free. Different examples illustrate the abstract concepts, special focus is on surface based coupling techniques and highly non-linear dimension reduced transport systems.