The Weight-Dependent Random Connection Model combines long-range percolation with scale-free network models. The talk focuses on the "weak decay regime" where connection probability tails are heavy enough to circumvent many geometrical difficulties that arise in short-range percolation models in low dimensions. I will summarise known sufficient conditions for existence and transience of an infinite component and discuss a new local existence theorem which improves upon a result of Berger (2002) for homogeneous long-range percolation and which implies the most general sufficient condition for transience hitherto known. A further pleasant consequence of the result is the continuity of the percolation function throughout the weak decay regime in dimension at least two.