This paper introduces a novel family of devices for ultra-fast, ultra-sensitive and ultra-low-noise optoelectronics, namely superconductive traveling-wave photodetectors (STWPDs). STWPDs can be optimized for ultra-wideband/ultra-sensitive photodetection, photomixing and microwave/THz waveform generation. An STWPD is a distributed device, both electrically and optically, which consists of a superconductive thin film kinetic-inductive photodetector integrated with a microwave/THz transmission line and an optical waveguide. We will present the principles of operation of STWPDs, a framework to calculate the input-output relation and the effects of the modal characteristics of the optical propagation on the device performance. Moreover, we will elaborate on how one can design an STWPD for each of the aforementioned applications. Different strategies for the design of the optical waveguide as well as some important design rules will also be discussed. We will report the current status of STWPDs and point out some key issues that should be met in the future for further development.