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We generalize and unify the $f(R,T)$ and $f(R,L_m)$ type gravity models byassuming that the gravitational Lagrangian is given by an arbitrary function ofthe Ricci scalar $R$, of the trace of the energy-momentum tensor $T$, and ofthe matter Lagrangian $L_m$, so that $L_{grav}=f(R,L_m,T)$. We obtain thegravitational field equations in the metric formalism, the equations of motionfor test particles, and the energy and momentum balance equations, which followfrom the covariant divergence of the energy-momentum tensor. Generally, themotion is non-geodesic, and takes place in the presence of an extra forceorthogonal to the four-velocity. The Newtonian limit of the equations of motionis also investigated, and the expression of the extra acceleration is obtainedfor small velocities and weak gravitational fields. The generalized Poissonequation is also obtained in the Newtonian limit, and the Dolgov-Kawasakiinstability is also investigated. The cosmological implications of the theoryare investigated for a homogeneous, isotropic and flat Universe for twoparticular choices of the Lagrangian density $f(R,L_m,T)$ of the gravitationalfield, with a multiplicative and additive algebraic structure in the mattercouplings, respectively, and for two choices of the matter Lagrangian, by usingboth analytical and numerical methods.

Generalizing the coupling between geometry and matter: $f(R,L_m,T)$ gravity