Multiscale entropy (MSE) is a widely-used tool to analyze biomedical signals. It was proposed to overcome the deficiencies of conventional entropy methods when quantifying the complexity of time series. However, MSE is undefined for very short signals and slow for real-time applications because of the use of sample entropy (SampEn). To overcome these shortcomings, we introduce multiscale dispersion entropy (DisEn - MDE) as a very fast and powerful method to quantify the complexity of signals. MDE is based on our recently developed DisEn, which has a computation cost of O(N), compared with O(N2) for SampEn. We also propose the refined composite MDE (RCMDE) to improve the stability of MDE. We evaluate MDE, RCMDE, and refined composite MSE (RCMSE) on synthetic signals and find that these methods have similar behaviors but the MDE and RCMDE are significantly faster than MSE and RCMSE, respectively. The results also illustrate that RCMDE is more stable than MDE for short and noisy signals, which are common in biomedical applications. To evaluate the proposed methods on real signals, we employ three biomedical datasets, including focal and non-focal electroencephalograms (EEGs), blood pressure recordings in Fantasia database, and resting-state EEGs activity in Alzheimer's disease (AD). The results again demonstrate a similar behavior of RCMSE, MDE and RCMDE, although the RCMDE and MDE are significantly faster and lead to larger differences between physiological conditions known to alter the complexity of the physiological recordings. To sum up, MDE and RCMDE are expected to be useful for the analysis of physiological signals thanks to their ability to distinguish different types of dynamics.