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This paper presents two new trajectory probability hypothesis density (TPHD) and trajectory cardinality probability hypothesis density (TCPHD) filters for joint tracking and classification (JTC), namely JTC-TPHD and JTC-TCPHD filters. We first introduce the classified trajectory RFS model to accommodate the motion model-based class information. The adaptation of the TPHD and TCPHD filters to the classified trajectory RFS is then formulated, i.e., JTC-TPHD and JTC-TCPHD, which are capable of jointly estimating the number, class, motion model, and trajectory status of multiple targets. We also develop the linear Gaussian implementation (LGM) as an analytic closed-form solution of the proposed filters. Simulation experiments discuss the validity and superiority of the proposed filters for multi-target JTC. • The classified trajectory variable to accommodate class information. • The proposal of JTC-T(C)PHD filters for joint tracking and classification. • The development of linear Gaussian mixture implementations. • The simulation demonstration of algorithms' validity and superiority. [ABSTRACT FROM AUTHOR]