We propose a scheme to reduce readout errors in experiments on quantum systems with finite number of measurement outcomes. Our method relies on performing classical post-processing which is preceded by Quantum Detector Tomography, i.e., the reconstruction of a Positive-Operator Valued Measure (POVM) describing the given quantum measurement device. If the measurement device is affected only by an invertible classical noise, it is possible to correct the outcome statistics of future experiments performed on the same device. To support the practical applicability of this scheme for near-term quantum devices, we characterize measurements implemented in IBM's and Rigetti's quantum processors. We find that for these devices, based on superconducting transmon qubits, classical noise is indeed the dominant source of readout errors. Moreover, we analyze the influence of the presence of coherent errors and finite statistics on the performance of our error-mitigation procedure. Applying our scheme on the IBM's 5-qubit device, we observe a significant improvement of the results of a number tasks including Quantum State Tomography, Quantum Process Tomography, the implementation of non-projective measurements, and certain quantum algorithms, such as Grover's search, the Bernstein-Vazirani algorithm. For more advanced variational quantum algorithms executed on a 15-qubit IBM device, we introduce a correlated measurement noise model that can be efficiently parametrized and which admits noise mitigation on the level of marginal probability distributions. Characterization of the noise model can be done efficiently using a generalization of the Wilczek Quantum Overlapping Tomography. We observe numerically that for numerous variational quantum optimization algorithms the noise-mitigation radically improves the quality of the optimization.