For nearly 50 years, the vision of using single molecules in circuits has been seen as providing the ultimate miniaturization of electronic chips. An advanced example of such a molecular electronics chip is presented here, with the important distinction that the molecular circuit elements play the role of general-purpose single-molecule sensors. The device consists of a semiconductor chip with a scalable array architecture. Each array element contains a synthetic molecular wire assembled to span nanoelectrodes in a current monitoring circuit. A central conjugation site is used to attach a single probe molecule that defines the target of the sensor. The chip digitizes the resulting picoamp-scale current-versus-time readout from each sensor element of the array at a rate of 1,000 frames per second. This provides detailed electrical signatures of the single-molecule interactions between the probe and targets present in a solution-phase test sample. This platform is used to measure the interaction kinetics of single molecules, without the use of labels, in a massively parallel fashion. To demonstrate broad applicability, examples are shown for probe molecule binding, including DNA oligos, aptamers, antibodies, and antigens, and the activity of enzymes relevant to diagnostics and sequencing, including a CRISPR/Cas enzyme binding a target DNA, and a DNA polymerase enzyme incorporating nucleotides as it copies a DNA template. All of these applications are accomplished with high sensitivity and resolution, on a manufacturable, scalable, all-electronic semiconductor chip device, thereby bringing the power of modern chips to these diverse areas of biosensing.