A new ESPRIT-based algorithm is proposed to estimate the direction-of-arrival of an arbitrary degree polynomial-phase signal with a single acoustic vector sensor. The proposed approach requires neither a priori knowledge of the polynomial-phase signal's coefficients nor a priori knowledge of the polynomial-phase signal's frequency-spectrum. A pre-processing technique is also proposed to incorporate the single-forgetting-factor algorithm and multiple-forgetting-factor adaptive tracking algorithm to track a polynomial-phase signal using one acoustic vector sensor. Simulation results verify the efficacy of the proposed direction finding and source tracking algorithms.