We propose LETO, a new hybrid Lagrangian-Eulerian method for topology optimization. At the heart of LETO lies in a hybrid particle-grid Material Point Method (MPM) to solve for elastic force equilibrium. LETO transfers density information from freely movable Lagrangian carrier particles to a fixed set of Eulerian quadrature points. The quadrature points act as MPM particles embedded in a lower-resolution grid and enable sub-cell resolution of intricate structures with areduced computational cost. By treating both densities and positions of the carrier particles as optimization variables, LETO reparameterizes the Eulerian solution space of topology optimizationin a Lagrangian view. LETO also unifies the treatment for both linear and non-linear elastic materials. In the non-linear deformation regime, the resulting scheme naturally permits large deformation and buckling behaviors. Additionally, LETO explores contact-awareness during optimization by incorporating a fictitious domain-based contact model into the static equilibrium solver, resulting in the discovery of novel structures. We conduct an extensive set of experiments. By comparing against a representative Eulerian scheme, LETO’s objective achieves an average quantitative improvement of 20% (up to 40%) in 3D and 2% in 2D (up to 12%). Qualitatively, LETO also discovers novelnon-linear functional structures and conducts self-contact-aware structural explorations.