The very high frame rate afforded by ultrafast ultrasound, combined with microbubble contrast agents, opens new opportunities for imaging tissue microvasculature. However, new imaging paradigms are required to obtain superior image quality from the large amount of acquired data while allowing real-time implementation. In this paper, we report a technique - acoustic sub-aperture processing (ASAP) - capable of generating very high contrast/SNR images of macro- and microvessels, with similar computational complexity to classical Power Doppler (PD) imaging. In ASAP, the received data are split into sub- groups. The reconstructed data from each sub-group are temporally correlated over frames to generate the final image. As signals in sub-groups are correlated but the noise is not, this substantially reduces the noise floor compared to PD. Using a clinical imaging probe, the method is shown to visualize vessels down to 200μm with a SNR 10dB higher than PD, and to resolve microvascular flow/perfusion information in rabbit kidneys non-invasively in vivo at multiple centimeter depth. With careful filter design, the technique also allows estimation of flow direction and separation of fast flow from tissue perfusion. ASAP can readily be implemented into hardware/firmware for real-timing imaging, and can be applied to contrast enhanced and potentially non-contrast imaging and 3D imaging.