Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett, Garcke, and N\"urnberg (J. Comput. Phys., 222 (2007), pp.~441--467), due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to first-order accuracy in time, and how to develop a higher-order numerical scheme is challenging. In this talk, we discuss the fully discrete, temporal higher-order parametric finite element methods for solving geometric flows of curves. The scheme are constructed based on the BGN formulation, a linear finite element approximation in space and various time stepping discretizations including Crank-Nicolson leap-frog method, Predictor-corrector method and backward difference formulae method. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGN-based scheme is higher-order accurate in time in terms of shape metrics. Moreover our proposed higher-order schemes exhibit good properties with respect to the mesh distribution. This talk is based on a joint work with Wei Jiang and Chunmei Su.