The human brain is a complex, integrated system defined by higher-order, multi-way interactions that are inherently simplified by traditional pairwise network models. Although a diverse array of analytical methods has been developed to address this limitation, the literature remains fragmented, lacking a unifying conceptual framework to organize these emerging developments. This review provides a comprehensive synthesis of higher-order functional connectivity (HOFC) methodologies in brain connectomics. We propose a fundamental distinction between two complementary paradigms based on their analytical outputs: Topographical HOFC and Temporal HOFC. We trace the evolution of Topographical HOFC from foundational Correlation-of-Correlations (CoC) and spatial profile similarity measures to advanced multi-view fusion frameworks and graph neural networks. Concurrently, we survey Temporal HOFC, exploring dynamic sliding-window clustering, meta-connectivity, high-resolution edge-centric models, and statistically rigorous, model-based approaches like the Matrix Variate Normal Distribution (MVND). By critically examining the conceptual, statistical, and computational challenges of these frameworks—including the biological interpretation gap, the "genuineness" of observed interactions, and the curse of dimensionality—we outline key comparative insights. Ultimately, we argue that the future of the field lies in a principled, neurobiologically grounded integration of these spatial and temporal approaches to unlock a robust, multi-level understanding of complex brain organization in health and disease.