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Frequency offsets negatively impact the performance of closed-loop multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems. Particularly, when multiple users are active, the impact can be high. Linear precoding and non-linear Tomlinson-Harashima precoding (THP) are thus developed for spatially-multiplexed multiuser OFDM and orthogonal space-time block-coded (OSTBC) OFDM. The proposed precoders employ a limited feedback structure, which is implemented with a shared codebook of precoding matrices, and only the index of the selected optimal matrix is fed back to the transmitter. The conventional limited feedback design criterion for flat-fading MIMO channels is only applicable to single-user OFDM without frequency offsets. We show that the ICI matrix due to frequency offset does not impact users precoding individually, and precoding on a per-subcarrier basis is possible. Exploiting this property, the conventional design is generalized to multiuser OFDM with frequency offsets. Nonlinear precoding uses a modulo arithmetic precoding matrix (which reduces the power efficiency loss inherent in linear precoding and leads to a lower error rate) and outperforms linear precoding. Our precoders not only offer significant bit error rate (BER) improvement for spatially-multiplexed multiuser MIMO OFDM with frequency offsets, but are equally effective for both OSTBC MIMO OFDM and spatially correlated channels.