The objective of this contribution is to elaborate on the notion of “traction continuity” across an interface at finite deformations. The term interface corresponds to a zero-thickness model representing the interphase between different constituents in a material. Commonly accepted interface models are the cohesive interface model and the elastic interface model. Both the cohesive and elastic interface models are the limit cases of a generalized interface model. This contribution aims to rigorously analyze the concept of the traction jump for the general interface model. The governing equations of the general interface model in the material as well as spatial configurations are derived and the traction jump across the interface for each configuration is highlighted. It is clearly shown that the elastic interface model undergoes a traction jump in both the material and spatial configurations according to a generalized Young-Laplace equation. For the cohesive interface model, however, while the traction field remains continuous in the material configuration, it can suffer a jump in the spatial configuration. This finding is particularly important since the cohesive interface model is based on the assumption of traction continuity across the interface and that the term “traction” often refers to the spatial configuration and not the material one. Thus, additional care should be taken when formulating an interface model in a geometrically non-linear framework. The theoretical findings for various interface models are carefully illustrated via a series of two-dimensional and three-dimensional numerical examples using the finite element method.