The concept of hybrid high-voltage alternating current (HVAC) and high-voltage direct current (HVDC) grid systems brings a massive advantage to reduce AC line loading, increased utilization of network infrastructure, and lower operational costs. However, it comes with issues, such as integration challenges, control strategies, optimization control, and security. The combined objectives in hybrid HVAC–HVDC grids are to achieve the fast regulation of DC voltage and frequency, optimal power flow, and stable operation during normal and abnormal conditions. The rise in hybrid HVAC–HVDC grids and associated issues are reviewed in this study along with state-of-the-art literature and developments that focus on modeling robust droop control, load frequency control, and DC voltage regulation techniques. The definitions, characteristics, and classifications of key issues are introduced. The paper summaries the key insights of hybrid HVAC–HVDC grids, current developments, and future research directions and prospects, which have led to the evolution of this field. Therefore, the motivation, novelty, and the main contribution of the survey is to comprehensively analyze the integration challenges, implemented control algorithms, employed optimization algorithms, and major security challenges of hybrid HVAC–HVDC systems. Moreover, future research prospects are identified, such as security algorithms’ constraints, dynamic contingency modeling, and cost-effective and reliable operation.