The realization of microscale total analysis systems and lab-on-a-chip technologies requires efficient actuation (mixing, pumping, atomizing, nebulizing, driving, etc.) of fluids on the microscopic scale and dexterous manipulation (separation, sorting, trapping, concentration, merging, patterning, aligning, focusing, etc.) of micro-objects (cells, droplets, particles, nanotubes, etc.) in open (sessile droplets) as well as confined spaces (microchannels/chambers). These capabilities have been recently achieved using powerful acoustofluidic techniques based on high-frequency (10-1000 MHz) surface acoustic waves (SAWs). SAW-based miniaturized microfluidic devices are best known for their non-invasive properties, low costs, and ability to manipulate micro-objects in a label-free manner. The energy-efficient SAWs are also compatible with conventional microfabrication technologies. The present work critically analyses recent reports describing the use of SAWs in microfluidic actuation and micro-object manipulation. Acoustofluidic techniques may be categorized according to the use of travelling SAWs (TSAWs) or standing SAWs (SSAWs). TSAWs are used to actuate fluids and manipulate micro-objects via acoustic streaming flow (ASF) as well as acoustic radiation force (ARF). SSAWs are mainly used for micro-object manipulation and are rarely employed for microfluidic actuation. We have reviewed reports of new technological developments that have not been covered in other recent reviews. In the end, we describe the future prospects of SAW-based acoustofluidic technologies.