Superconducting nanowire single-photon detectors with peak efficiencies above 90% and unrivalled timing jitter (<30 ps) have emerged as a potent technology for quantum information and sensing applications. However, their high cost and cryogenic operation limit their widespread applicability. Here, we present an approach using tapered InP nanowire p-n junction arrays for high-efficiency, broadband and high-speed photodetection without the need for cryogenic cooling. The truncated conical nanowire shape enables a broadband, linear photoresponse in the ultraviolet to near-infrared range (similar to 500 nm bandwidth) with external quantum efficiencies exceeding 85%. The devices exhibit a high gain beyond 10(5), such that a single photon per pulse can be distinguished from the dark noise, while simultaneously showing a fast pulse rise time (<1ns) and excellent timing jitter (<20 ps). Such detectors open up new possibilities for applications in remote sensing, dose monitoring for cancer treatment, three-dimensional imaging and quantum communication.