We present the model, design, and implementation of SMARS, the first practical Sleep Monitoring system that exploits Ambient Radio Signals to recognize sleep stages and assess sleep quality. This will enable a future smart home that monitors daily sleep in a ubiquitous, non-invasive and contactless manner, without instrumenting the subject's body or the bed. The key enabler underlying SMARS is a statistical model that accounts for all reflecting and scattering multipaths, allowing highly accurate and instantaneous breathing estimation with best-ever performance achieved on commodity devices. On this basis, SMARS then recognizes different sleep stages, including wake, rapid eye movement (REM), and non-REM (NREM), which was previously only possible with dedicated hardware. We implement a real-time system on commercial WiFi chipsets and deploy it in 6 homes, resulting in 32 nights of data in total. Our results demonstrate that SMARS yields a median absolute error of 0.47 breaths per minute (BPM) and a 95 percent-tile error of only 2.92 BPM for breathing estimation, and detects breathing robustly even when a person is 10 meters away from the link, or behind a wall. SMARS achieves a sleep staging accuracy of 88 percent, outperforming the prevalent unobtrusive commodity solutions using bed sensor or UWB radar. The performance is also validated upon a public sleep dataset of 20 patients. By achieving promising results with merely a single commodity RF link, we believe that SMARS will set the stage for a practical in-home sleep monitoring solution.