Over the past few decades, wind energy has expanded to become a widespread, clean, and sustainable energy source. However, integrating offshore wind energy with the onshore AC grids presents many stability and control challenges that hinder the reliability and resilience of AC grids, particularly during faults. To address this issue, current grid codes require offshore wind farms (OWFs) to remain connected during and after faults. This requirement is challenging because, depending on the fault location and power flow direction, DC link over- or under-voltage can occur, potentially leading to the shutdown of converter stations. Therefore, this necessitates the proper understanding of key technical concepts associated with the integration of OWFs. To help fill the gap, this article performs an in-depth investigation of existing alternating current fault ride through (ACFRT) techniques of modular multilevel converter-based high-voltage direct current (MMC-HVDC) for OWFs. These techniques include the use of AC/DC choppers, flywheel energy storage devices (FESDs), power reduction strategies for OWFs, and energy optimization of the MMC. This article covers both scenarios of onshore and offshore AC faults. Given the importance of wind turbines (WTs) in transforming wind energy into mechanical energy, this article also presents an overview of four WT topologies. In addition, this article explores the advanced converter topologies employed in HVDC systems to transform three-phase AC voltages to DC voltages and vice versa at each terminal of the DC link. Finally, this article explores the key stability and control concepts, such as small signal stability and large disturbance stability, followed by future research trends in the development of converter topologies for HVDC transmission such as hybrid HVDC systems, which combine current source converters (CSCs) and voltage source converters (VSCs) and diode rectifier-based HVDC (DR-HVDC) systems.