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Millimeter-wave (mmWave) with large spectrum available is considered as the most promising frequency band for future wireless communications. The IEEE 802.11ad and IEEE 802.11ay operating on 60 GHz mmWave are the two most expected wireless local area network (WLAN) technologies for ultra-high-speed communications. For the IEEE 802.11ay standard still under development, there are plenty of proposals from companies and researchers who are involved with the IEEE 802.11ay task group. In this survey, we conduct a comprehensive review on the medium access control layer (MAC) related issues for the IEEE 802.11ay, some cross-layer between physical layer (PHY) and MAC technologies are also included. We start with MAC related technologies in the IEEE 802.11ad and discuss design challenges on mmWave communications, leading to some MAC related technologies for the IEEE 802.11ay. We then elaborate on important design issues for IEEE 802.11ay. Specifically, we review the channel bonding and aggregation for the IEEE 802.11ay, and point out the major differences between the two technologies. Then, we describe channel access and channel allocation in the IEEE 802.11ay, including spatial sharing and interference mitigation technologies. After that, we present an in-depth survey on beamforming training (BFT), beam tracking, single-user multiple-input-multiple-output (SU-MIMO) beamforming and multi-user multiple-input-multiple-output (MU-MIMO) beamforming. Finally, we discuss some open design issues and future research directions for mmWave WLANs. We hope that this paper provides a good introduction to this exciting research area for future wireless systems.