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In this proceeding, we review modified theories of gravity with acurvature-matter coupling between an arbitrary function of the scalar curvatureand the Lagrangian density of matter. This explicit nonminimal coupling inducesa non-vanishing covariant derivative of the energy-momentum tensor, thatimplies non-geodesic motion and consequently leads to the appearance of anextra force. Here, we explore the physical and cosmological implications of thenonconservation of the energy-momentum tensor by using the formalism ofirreversible thermodynamics of open systems in the presence of mattercreation/annihilation. The particle creation rates, pressure, and theexpression of the comoving entropy are obtained in a covariant formulation anddiscussed in detail. Applied together with the gravitational field equations,the thermodynamics of open systems lead to a generalization of the standard$\Lambda$CDM cosmological paradigm, in which the particle creation rates andpressures are effectively considered as components of the cosmological fluidenergy-momentum tensor. Furthermore, we also briefly present the coupling ofcurvature to geometry in conformal quadratic Weyl gravity, by assuming acoupling term of the form $L_m\tilde{R}^2$, where $L_m$ is the ordinary matterLagrangian, and $\tilde{R}$ is the Weyl scalar. The coupling explicitlysatisfies the requirement of the conformal invariance of the theory. Expressing$\tilde{R}^2$ with the use of an auxiliary scalar field and of the Weyl scalar,the gravitational action can be linearized in the Ricci scalar, leading in theRiemann space to a conformally invariant $f\left(R,L_m\right)$ type theory,with the matter Lagrangian nonminimally coupled to geometry.

Curvature-matter couplings in modified gravity: from linear models to conformally invariant theories