Representation-based classification (RC) methods such as sparse RC (SRC) have attracted great interest in pattern recognition recently. Despite their empirical success, few theoretical results are reported to justify their effectiveness. In this paper, we establish the theoretical guarantees for a general unified framework termed as atomic representation-based classification (ARC), which includes most RC methods as special cases. We introduce a new condition called atomic classification condition (ACC), which reveals important geometric insights for the theory of ARC. We show that under such condition ARC is provably effective in correctly recognizing any new test sample, even corrupted with noise. Our theoretical analysis significantly broadens the range of conditions under which RC methods succeed for classification in the following two aspects: (1) prior theoretical advances of RC are mainly concerned with the single SRC method while our theory can apply to the general unified ARC framework, including SRC and many other RC methods; and (2) previous works are confined to the analysis of noiseless test data while we provide theoretical guarantees for ARC using both noiseless and noisy test data. Numerical results are provided to validate and complement our theoretical analysis of ARC and its important special cases for both noiseless and noisy test data.