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An accurate estimation of the magnitudes of three in-situ stresses can usefully improve the production performance of hydraulic fracturing and reduce the risk of wellbore instability. Considering two types of linear elastic stress around the non-perforated arbitrarily inclined borehole (plane strain and plane stress conditions), the theory of hydraulic fracturing, and the maximum tensile criterion, the generalized governing formulas are derived to explain the mechanism of hydraulically or drilling-induced longitudinal, oblique, and transverse tensile wall-fractures in arbitrarily inclined boreholes. These formulas provide useful implications for constraints on the magnitudes of in-situ stresses, which determine the initiation of different types of tensile wall-fractures under different faulting stress regimes. The present paper extends the initiating conditions of the common transverse tensile wall-fractures related to the works of Hossain et al. (2000) and Nelson et al. (2005). The prerequisite for the special transverse tensile wall-fractures that tensile fractures simultaneously initiated around a whole borehole wall (Huang et al., 2012) is generalized by introducing the concept of the turning locations. The formulas corresponding to the turning locations are provided under different faulting stress regimes, revealing that the work proposed by Huang et al. (2012) is only a particular case. It is found that the turning locations are related to the critical borehole inclination or azimuth angles. Overall, all the formulas in this study for longitudinal, oblique, and transverse tensile wall-fractures can be used to design hydraulic fracturing experiments and to determine the in-situ stress by using the information of drilling-induced tensile wall-fractures. Moreover, the governing formulas of the turning locations can also be used to optimize the borehole trajectory. • Potential candidates of transverse hydraulically induced or drilling-induced tensile wall-fractures are generally formulated. • The same initiating sequences for longitudinal tensile wall-fractures occur between the vertical and horizontal boreholes. • The governing formulas of transverse and longitudinal tensile wall-fractures constrain the magnitudes of in-situ stresses. [ABSTRACT FROM AUTHOR]