Novel devices based on orthorhombic κ-Ga2O3 could enable solar blind infrared detection or high-electron mobility transistors with large two-dimensional electron gas densities. Here, we report on the current transport parallel to the growth direction of κ-Ga2O3 layers grown by pulsed-laser deposition on highly conductive Al-doped ZnO back contact layers. Besides ohmic Ti/Al/Au contact layer stacks, vertical Pt/PtOx/κ-Ga2O3 and Pd/PdOx/κ-Ga2O3 Schottky barrier diodes and NiO/κ-Ga2O3 and ZnCo2O4/κ-Ga2O3 pn-heterodiodes are investigated by current–voltage measurements. While a lateral current transport is severely suppressed to less than 10−9Acm−2 due to rotational domains, we record a significant current flow through the ohmic contacts in the vertical direction of >0.1Acm−2. The Schottky barrier diodes and the pn-heterojunctions exhibit rectification ratios of up to seven orders of magnitude. Room temperature current–voltage characteristics of diode ensembles as well as temperature-dependent measurements for selected Pt-based diodes reveal a mean barrier height of ϕBm≈2.1eV and ideality factors down to η≈1.3.