Abstract: The well-known generalized-$\alpha$ method is an unconditionally stable and second-order accurate time-integrator which has a feature of user-control on numerical dissipation. The method encompasses a wide range of time-integrators, such as the Newmark method, the HHT-$\alpha$ method by Hilber, Hughes, and Taylor, and the WBZ-$\alpha$ method by Wood, Bossak, and Zienkiewicz. The talk starts with the simplest time-integrator, forward/backward Euler schemes, then introduces Newmark's idea followed by the ideas of Chung and Hulbert on the generalized-$\alpha$ method. The focus of the talk is to introduce two ideas to generalize the method further to higher orders while maintaining the features of unconditional stability and dissipation control. We will show third-order and $2n$-order accurate schemes with numerical validations. The talk closes with the introduction of a variational-splitting framework for these time-integrators. As a consequence, the splitting schemes reduce the computational costs significantly (to linear cost) for multi-dimensional problems. Collaborating authors: Pouria Behnoudfar, Victor Calo, Alexandre Ern, Peter Minev, Maciej Paszy\'{n}ski.