<p>Classical continuum theory, grounded in the scale separation hypothesis, is inadequate for accurately capturing size-dependent mechanical responses in micro/nano-scale structures. In contrast, couple stress theory (CST), a representative higher-order continuum theory, addresses this limitation by introducing additional material length scale parameters to characterize size effects, enabling efficient solutions within the continuum mechanics framework. However, the numerical implementation of CST, essential for practical applications, confronts two critical challenges. The first is the requirement of interelement <i>C</i>¹ continuity, namely, the continuity of both displacements and mechanical rotations, which significantly complicates element construction and induces high sensitivity to mesh distortion. Unlike isogeometric analysis (IGA), which inherently satisfies high-order continuity requirements, conventional low-order finite elements struggle to meet these demands, particularly for two/three-dimensional geometries. The second challenge lies in the incomplete formulation of geometrically nonlinear CST, especially regarding ambiguous rate measures for curvature and couple-stress in three-dimensional problems. Over recent decades, significant effort has been devoted to developing conventional low-order finite elements based on CST. This paper comprehensively reviews these contributions, with a particular focus on examining aforementioned two key challenges and future research directions. Addressing these challenges is crucial for advancing the robust modeling of size-dependent behaviors in micro/nano-devices using the CST-based FEM.</p>