Nanozymes are nanomaterials with enzyme-like activities. Compared to natural enzymes, nanozymes are more stable and cost-effective, and they have unique properties due to their nanoscale size and surface chemistry. In this review, we summarize 'signal-on' nanozyme-based sensors for detecting metal ions, anions, small molecules and proteins. Since protein-based enzymes are already highly active, they were used to detect their inhibitors, resulting in 'signal-off' sensors. On the other hand, for nanozymes, target molecules were detected either as a promotor of nanozyme activity or for its ability to selectively remove nanozyme inhibitors. In both cases, 'signal-on' detection was achieved. We classify the commonly used nanozymes based on their composition such as metal oxide, gold nanoparticles and other nanomaterials, most of which belong to the oxidase, peroxidase and catalase mimics. The nanozymes can catalyze the oxidation of colorless or non-fluorescent substrates to produce a visual or fluorescent signal. Based on this, this article presents some typical 'turn-on' and 'turn-off-on' sensors, and we critically review their design principles. At the end, further perspectives for the nanozyme-based sensors are outlined.