This paper introduces heart sound detection by radar systems, which enables touch-free and continuous monitoring of heart sounds. The proposed measurement principle entails two enhancements in modern vital sign monitoring. First, common touch-based auscultation with a phonocardiograph can be simplified by using biomedical radar systems. Second, detecting heart sounds offers a further feasibility in radar-based heartbeat monitoring. To analyse the performance of the proposed measurement principle, 9930 seconds of eleven persons-under-tests' vital signs were acquired and stored in a database using multiple, synchronised sensors: a continuous wave radar system, a phonocardiograph (PCG), an electrocardiograph (ECG), and a temperature-based respiration sensor. A hidden semi-Markov model is utilised to detect the heart sounds in the phonocardiograph and radar data and additionally, an advanced template matching (ATM) algorithm is used for state-of-the-art radar-based heartbeat detection. The feasibility of the proposed measurement principle is shown by a morphology analysis between the data acquired by radar and PCG for the dominant heart sounds S1 and S2: The correlation is 82.97 ± 11.15% for 5274 used occurrences of S1 and 80.72 ± 12.16% for 5277 used occurrences of S2. The performance of the proposed detection method is evaluated by comparing the F-scores for radar and PCG-based heart sound detection with ECG as reference: Achieving an F1 value of 92.22 ± 2.07%, the radar system approximates the score of 94.15 ± 1.61% for the PCG. The accuracy regarding the detection timing of heartbeat occurrences is analysed by means of the root-mean-square error: In comparison to the ATM algorithm (144.9 ms) and the PCG-based variant (59.4 ms), the proposed method has the lowest error value (44.2 ms). Based on these results, utilising the detected heart sounds considerably improves radar-based heartbeat monitoring, while the achieved performance is also competitive to phonocardiography.