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In the field of radar, nonlinearity has always been significant challenge in target tracking algorithms. It is evident in the complexity of the target motion model, observation model, and maneuverability of the target. Traditional model-based algorithms often rely on numerical approximations or simulations to obtain suboptimal solutions, which may lead to conversion errors and increase algorithm complexity. Model-free methods based on deep neural networks (DNNs) have been continuously employed in nonlinear target tracking (NTT) to improve target state estimation performance. This article introduces two nonlinear trackers based on the Transformer that are used for smoothing, filtering, and predicting target states in the NTT task. First, a classical Transformer-based method is proposed for smoothing and prediction, improving both inference efficiency and accuracy through parallel operation. After that, to handle the recursive operation required for filtering, we introduce a novel recursive Transformer for recursive filtering and predicting of the target state. This significantly reduces computational load compared to the classical Transformer method. Simulation results indicate that the proposed algorithm outperforms traditional and recurrent neural network (RNN)-based methods.