Detecting nonclassical light is a central requirement for photonics-based quantum technologies. Unrivaled high efficiencies and low dark counts have positioned superconducting nanowire single-photon detectors (SNSPDs) as the leading detector technology for integrated photonic applications. However, a central challenge lies in their integration within photonic integrated circuits, regardless of material platform or surface topography. Here, we introduce a method based on transfer printing that overcomes these constraints and allows for the integration of SNSPDs onto arbitrary photonic substrates. With a kinetically controlled elastomer stamp, we transfer suspended SNSPDs onto commercially manufactured silicon and lithium niobate on insulator integrated photonic circuits. Focused ion beam metal deposition then wires the detectors to the circuits, thereby allowing us to monitor photon counts with >7% detection efficiencies. Our method eliminates detector integration bottlenecks and provides new venues for versatile, accessible, and scalable quantum information processors.