Previous theoretical formulations for sinusoidal and helical buckling of drillstrings vary significantly and are mostly proposed for frictionless pipes without tool joints. Finite-element-analysis (FEA) methods have the ability to consider geometric details and nonlinearities. However, traditional FEA methods use shell or solid elements for this problem and are computationally expensive. In this paper, an explicit FEA that is based on beam and connector elements implemented in the Abaqus software is used to study the problem in different wellbores. The wellbore geometry, formation stiffness, friction load, and friction-induced torque are modeled with connector elements. A typical drillstring in vertical, inclined, horizontal, and curved wellbores is simulated, and the explicit FEA results for sinusoidal- and helical-buckling loads are compared with different theoretical formulations and experimental results in the literature. The effects of length, inclination angle, and string effective weight caused by buoyancy as well as the effect of tool joints in straight and curved wellbores are also studied and compared with present formulations and published experimental results. A full-scale well-buckling test has also been simulated, and load-transfer data are compared with the real-world results. Overall, it is demonstrated that the use of explicit FEA can efficiently study drillstring buckling behavior in straight- and curved-wellbore conditions.