This report describes a particle tracking computer program named PT123. The development of PT123 was supported in part by the Civil Works Basic Research project entitled "Efficient Resolution of Complex Transport Phenomena Using Eulerian-Lagrangian Techniques" and in part by the System-Wide Water Resources Program (SWWRP). Given velocities, PT123 can track massless particles in 1-, 2-, and 3-D unstructured or converted structured meshes. The elements used to construct PT123 meshes are line elements in 1-D, triangular and/or quadrilateral elements in 2-D, and tetrahedral, triangular prism, and/or hexahedral elements in 3-D. One adaptive (embedded 4th-and 5th-order) and three non-adaptive (1st-, 2nd-, and 4th-order) Runge-Kutta (RK) methods are included in PT123 to solve the ordinary differential equations describing the motion of particles. The adaptive RK method allows the user to control tracking accuracy with specified error tolerances. The nonadaptive RK methods provide the user options to balance computational efficiency and accuracy by using lower order schemes for smooth velocity fields and higher order schemes for complex velocity fields. Both elementby-element (EBE) and non-element-by-element (NEBE) tracking approaches are incorporated into PT123. Both node-and element-based velocity can be used for particle tracking. PT123 can execute forward and backward tracking and output tracking history at a specified frequency. It tracks particles along the closed boundary and stops tracking when a particle encounters the open boundary through which particles enter or exit the computational domain. The start and end times of tracking are flexible as long as their corresponding velocities can be computed via temporal interpolation using the given velocities. This report is the first report of the series describing the development and application of PT123. It details the governing equation and numerical approaching associated with PT123 Version 1.0. Six test examples in multiple dimensions are used for verification and demonstration. The structure and the input guide of the computer program are given in the appendices. DISCLAIMER:The contents of this report are not to be used for advertising, publication, or promotional purposes. Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products. All product names and trademarks cited are the property of their respective owners. The findings of this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents.DESTROY THIS REPORT WHEN NO LONGER NEEDED. DO NOT RETURN IT TO THE ORIGINATOR.