An accurate singularity-free geometrically exact formulation of a three-dimensional beam with large displacements, large deformations, and large rotations is developed. The undeformed configuration of the beam can be either straight or curved. Euler parameters are used to parametrize the rotation of a cross section of the beam to avoid singularity. While the position vector is interpolated using Hermite functions, Euler parameters are interpolated using a C 1 -continuous interpolation function. The stretch–shear strain vector of the centroid line of the beam can be obtained by independent interpolations of position and rotation fields. Governing equations of the beam are obtained using Lagrange’s equations for systems with constraints. An arc-length method is introduced to trace the force–displacement curve of equilibria of the beam. Several examples are simulated to show the performance of the current formulation. A static planar example with its exact solution is first simulated to show accuracy and convergency of the current formulation. In-plane buckling of a circular arch is then studied, and the snap-through phenomenon is found when the arch has clamped–clamped boundary conditions. The stiffness, natural frequencies, and mode shapes of a spring are calculated; the effect of the mass of the spring on natural frequencies and mode shapes of the spring–mass system is analyzed, and a curve veering phenomenon accompanied with mode shift and mode localization behaviors is found there. The current formulation is compared with the formulation in the commercial software ADAMS using a dynamic three-dimensional example, and their results are in excellent agreement. Another dynamic example of a moving force on a beam is also shown, and its dynamic behavior is analyzed.