Integrated sensors are essential for scanning probe microscopy (SPM) based systems that employ arrays of microcantilevers for high throughput. Common integrated sensors, such as piezoresistive, piezoelectric, capacitive and thermoelectric sensors, suffer from low bandwidth and/or low resolution. In this paper, a novel magnetoresistive-sensor-based scanning probe microscopy (MR-SPM) technique is presented. The principle of MR-SPM is first demonstrated using experiments with magnetic cantilevers and commercial MR sensors. A new cantilever design tailored to MR-SPM is then presented and micromagnetic simulations are employed to evaluate the achievable resolution. A remarkable resolution of 0.84 Å over a bandwidth of 1 MHz is estimated, which would significantly outperform state-of-the-art optical deflection sensors. Due to its combination of high resolution at high bandwidth, and its amenability to integration in probe arrays, MR-SPM holds great promise for low-cost, high-throughput SPM.