Two-dimensional N=(0, 1) and (0, 2) supersymmetric sigma models can be mainly obtained in two ways: non-minimal heterotic deformation of N=(1, 1) and (2, 2) sigma models, and minimal construction which contains only (0, 1) and (0, 2) supermultiplets. The former deformed models with N=(0, 2) supersymmetries emerge as low-energy world sheet theories on non-Abelian strings supported in some N=1 four-dimensional Yang-Mills theories. The latter, on the other hand, can be regarded as the elementary building blocks to construct generic N=(0, 1) and (0, 2) chiral models. In the thesis, we will study both types of sigma models. We start with the deformed heterotic sigma models with N=(0, 2) supersymmetries. Our investigation is around the calculation of NSVZ exact beta-function of the heterotic models through instanton technique, and also verifies it by straightforward two-loop calculation and the ``Konishi anomaly'' of the hypercurrent. Finally, we also consider isometries on their target spaces, and show that the heterotic deformation is free of isometry and holonomy anomalies. Then we turn to analysis of a more fundamental minimal construction of chiral sigma models with N=(0, 1) and (0, 2) supersymmetries. These minimal models with only (left) chiral fermions may intrinsically suffer from chiral anomalies that will render the theories mathematically inconsistent. We focus on two important examples, the minimal O(N) and CP(N-1) models, and calculate their isometry anomalies. We show that the CP(N-1) models with N>2 has non-removable chiral anomalies, while the O(N) models are anomaly free and thus exist quantum mechanically. We also disclose a relation between isometry anomalies in these non-linear sigma models (NLSM) and gauge anomalies in gauged linear sigma models (GLSM). Finally, we reveal a relation on anomaly correspondence between NLSM and GLSM to minimal models on homogeneous spaces. We interpret these anomalies more from geometric perspectives and relate them to the characteristic classes of the target spaces. Through explicit calculation of anomalous fermionic effective action, we show how to add a series of local counterterms to remove the anomalies. We eventually reach a result that the remedy procedure is equivalent to require the target spaces of theories with trivial first Pontryagin class, and thus demonstrate Moore and Nelson's consistency condition in the case of homogeneous spaces. More importantly, we find that local counterterms further constrain ``curable'' models and make some of them flow to non-trivial infrared superconformal fixed point. We also discuss a interesting relation between N=(0, 1) and (0, 2) supersymmetric sigma model and gauge theories in the spirit of 't Hooft anomaly matching condition.