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In this paper, we study the high-resolution channel estimation problem for intelligent reflecting surface (IRS)assisted millimeter wave (mmWave) multiple-input-multipleoutput (MIMO) communications, which is a prerequisite to guarantee further high-rate data transmission. Considering the typical sparsity of mmWave channels, we formulate the cascaded channel estimation problem from a sparse signal recovery perspective, and then propose a novel two-step cascaded channel estimation protocol to estimate the cascaded user-IRS-base station channel with high-resolution for IRS-assisted mmWave MIMO communications. More specifically, the first step is to estimate the coarse angular domain information (ADI) and further establish the robust uplink by beam training. In the second step, by exploiting the coarse ADI, an adaptive grid matching pursuit (AGMP) algorithm is proposed to estimate the high-resolution cascaded channel state information (CSI) with low complexity. Simulation results verify that the proposed two-step channel estimation protocol significantly outperforms the state-of-the-art scheme, i.e., beam training based channel estimation, and meanwhile can reap near-optimal system performance achieved by perfect CSI.