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With the increasing scale of antenna arrays in wideband millimeter-wave (mmWave) communications, the physical propagation delays of electromagnetic waves traveling across the whole array will become large and comparable to the time-domain sample period, which is known as the spatial-wideband effect. In this case, different subcarriers in an orthogonal frequency division multiplexing (OFDM) system will “see” distinct angles of arrival (AoAs) for the same path. This effect is known as beam squint, resulting from the spatial-wideband effect, and makes the approaches based on the conventional multiple-input multiple-output (MIMO) model, such as channel estimation and precoding, inapplicable. After discussing the relationship between beam squint and the spatial-wideband effect, we propose a channel estimation scheme for frequency-division duplex (FDD) mmWave massive MIMO-OFDM systems with hybrid analog/digital precoding, which takes the beam squint effect into consideration. A compressive sensing-based approach is developed to extract the frequency-insensitive parameters of each uplink channel path, i.e., the AoA and the time delay, and the frequency-sensitive parameter, i.e., the complex channel gain. With the help of the reciprocity of these frequency-insensitive parameters in FDD systems, the downlink channel estimation can be greatly simplified, where only limited pilots are needed to obtain downlink complex gains and reconstruct downlink channels. Furthermore, the uplink and downlink channel covariance matrices can be constructed from these frequency-insensitive channel parameters rather than through a long-term average, which enables the minimum mean-squared error (MMSE) channel estimation to further enhance performance. Numerical results demonstrate the superiority of the proposed scheme over the conventional methods under general system configurations in mmWave communications.