This work investigates a novel heat-driven multi-stage thermoacoustic cooler that can satisfy cooling requirements in the applications of natural gas liquefaction and hightemperature superconductivity. The proposed system consists of a compressor, multiple thermoacoustic units (engines and coolers) coupled by piston-cylinder assemblies. The acoustic power input by the compressor is successively multiplied in the thermoacoustic engine units, and the amplified acoustic power is then consumed to produce cooling power in the thermoacoustic cooler units. The proposed system overcomes the limitations of the traditional thermoacoustic systems owing to high efficiency, compact size, and ease of control. Analyses are first performed to explore the influence of the number of stages. The design method of the pistons is presented based on acoustic impedance matching principle. Based on the optimized system, simulations are then conducted to investigate the axial distribution of the key parameters, which can explain the reason for improved thermodynamic performance. At heating and cooling temperatures of 873 K and 130 K, the system achieves a cooling power of 2.1 kW and a thermal-to-cooling relative Carnot efficiency of 23%. This represents significant increases by over 60% in efficiency and 80% in cooling capacity when compared to existing systems. Simulations further demonstrate how controlling the input acoustic power and frequency via the compressor enables control of the system under various conditions. Further discussions are made considering a potential combined cooling and power system, indicating a thermal-cooling-electricity efficiency of 34% without any external electric power required for the compressor.