We investigate a large class of random graphs on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random weight and given weight and position of the points we form an edge between two points independently with a probability depending on the two weights and the distance of the points. This generalises many spatial random graph models. We investigate whether random walks on the infinite cluster is recurrent or transient. In dimensions \(d \in \{1,2\}\) we achieve a complete characterisation; for \(d \in 3\) we prove transience in all cases except for a regime where we conjecture that scale-free and long-range effects play no role. In a plain version of the random connection model, where weights are ignored, we can even analyse the model at the phase transition point. Indeed, we obtain an infrared bound for the critical connectivity function if the dimension is sufficiently large or if the pair connection function has sufficiently slow decay. This is achieved through an adaptation of the percolation lace expansion for Poisson processes. Based on joint work with Peter Gracar, Remco van der Hofstad, Günter Last, Kilian Matzke, Christian Mönch, and P. Mörters.