Electrical connectors for very high speed signals, including signals at or above 112 Gbps. Effectiveness of shielding along the signal paths through the mating electrical connectors may be enhanced through the use of one or more techniques, including enabling two-sided shielding, connections between shield members and between shield members and grounded structures of printed circuit boards to which the connectors are mounted, and selective positioning of lossy material. Such techniques may be simply and reliably implemented in high density connector using one or more techniques. An electrical connector may include core members held by a housing together with leadframe assemblies attached to the core members. The core members may include features that would be difficult to mold in a housing and may include both shields and lossy materials in locations that would be difficult to incorporate in a leadframe assembly.