SPHysics – development of a free-surface fluid solver – Part 1: Theory and formulations. A free-surface fluid solver called SPHysics is presented. Part 1 provides a description of the governing equations based on Smoothed Particle Hydrodynamics (SPH) theory. The paper describes the formulations implemented in the code including the classical SPH formulation along with enhancements like density filtering, arbitrary Lagrange–Euler (ALE) schemes and the incorporation of Riemann solvers for particle–particle interactions. Second-order time stepping schemes are presented along with the boundary conditions employed which can handle floating objects to study fluid–structure interaction. In addition, the model implementation is briefly described. This information will be used in Part 2, where the efficiency of the code is discussed, along with several study cases.