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Channel estimation is challenging for the reconfigurable intelligent surface (RIS)-aided wireless communications. Since the number of coefficients of the cascaded channel among the base station (BS), the RIS, and the user equipment (UE), is the product of the number of BS antennas, the number of RIS elements, and the number of UEs, the pilot overhead can be prohibitively high. In this paper, we propose a two-timescale channel estimation framework to exploit the property that the BS-RIS channel is high-dimensional but quasi-static, while the RIS-UE channel is mobile but low-dimensional. Specifically, to estimate the quasi-static BS-RIS channel, we propose a dual-link pilot transmission scheme, where the BS transmits downlink pilots and receives uplink pilots reflected by the RIS. Then, we propose a coordinate descent-based algorithm to recover the BS-RIS channel. Since the quasi-static BS-RIS channel is estimated less frequently than the mobile channel is, the average pilot overhead can be reduced from a long-term perspective. Although the mobile RIS-UE channel has to be frequently estimated in a small timescale, the associated pilot overhead is low thanks to its low dimension. Simulation results show that the proposed two-timescale channel estimation framework can achieve accurate channel estimation with low pilot overhead. [ABSTRACT FROM AUTHOR]