In this study, we delve into the robustness of neural network-based LiDAR point cloud tracking models under adversarial attacks, a critical aspect often overlooked in favor of performance enhancement. These models, despite incorporating advanced architectures like Transformer or Bird's Eye View (BEV), tend to neglect robustness in the face of challenges such as adversarial attacks, domain shifts, or data corruption. We instead focus on the robustness of the tracking models under the threat of adversarial attacks. We begin by establishing a unified framework for conducting adversarial attacks within the context of 3D object tracking, which allows us to thoroughly investigate both white-box and black-box attack strategies. For white-box attacks, we tailor specific loss functions to accommodate various tracking paradigms and extend existing methods such as FGSM, C\&W, and PGD to the point cloud domain. In addressing black-box attack scenarios, we introduce a novel transfer-based approach, the Target-aware Perturbation Generation (TAPG) algorithm, with the dual objectives of achieving high attack performance and maintaining low perceptibility. This method employs a heuristic strategy to enforce sparse attack constraints and utilizes random sub-vector factorization to bolster transferability. Our experimental findings reveal a significant vulnerability in advanced tracking methods when subjected to both black-box and white-box attacks, underscoring the necessity for incorporating robustness against adversarial attacks into the design of LiDAR point cloud tracking models. Notably, compared to existing methods, the TAPG also strikes an optimal balance between the effectiveness of the attack and the concealment of the perturbations.