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For applications, metallic glasses need to retain their high strength over enhanced strain ranges. However, many metallic glasses show catastrophic failure, even in or close to the end of the regime that conventionally has been thought to be elastic. Recent observations of irreversible events at low strains shed some doubt on this nomenclature. In fact, these observations indicate that although the macroscopic response indicates elastic behavior, the microscopic processes might at least partially be irreversible and time-dependent. In that respect, shear bands as the result of shear localization and as the cause of shear softening play a decisive role with respect to the performance of metallic glasses under mechanical load. Therefore, in this work, we are aiming at understanding the correlations between glass structure, glass properties and the thermomechanical history of the glass including the shear bands on one hand and the plastic yielding on the other hand. Various attempts have been made to explain or model the response of metallic glasses to externally applied shear stresses. Hypotheses have been put forward concerning the impact of materials parameters or structural aspects that might favor homogeneous plastic flow. Additionally, scenarios have been suggested for the early stages of anelastic and plastic deformation as well as for the transition to localization and shear band formation/activation. In this contribution, we present a focused viewpoint both from theory and modeling as well as an experimental perspective set on the structure and properties of glasses under shear, with a special focus on shear banding. We also discuss the impact of the local structure of glasses (that depend on the synthesis and processing path-way) in terms of their medium range order. The impact of chemical composition (including microalloying effects) on kinetic properties of shear bands and elasto-plastic properties (Poisson's ratio and three-point bending) is evaluated. These aspects have seen relatively sparse coverage, but are of far-reaching consequences concerning the properties of glasses under mechanical stress. Moreover, these aspects can reveal new insight into the underlying dominant mechanisms and, equally important, allow also to infer about the early stages of strain localization and to analyze the impact of structural heterogeneity or processing conditions on plastic yielding.