Let $(V,\omega)$ be a compact symplectic manifold which is simply connected and positive, such as Fano varieties in the Kähler case. According to Donaldson there is a category whose objects are the Lagrangian submanifolds with torsion fundamental group and whose morphisms are the Floer homology groups ${\rm Mor}(L_0,L_1)=HF^*(L_0,L_1)$. The composition maps form a product structure on Floer cohomology $$HF^*(L_0,L_1)\otimes HF^*(L_1,L_2)\to HF^*(L_0,L_2)$$ which can be defined in terms of holomorphic triangles. An example is where $V=\overline M\times M$ is a product manifold and the Lagrangian submanifolds are graphs of symplectomorphisms. This leads to a product structure $$HF^*(\phi)\otimes HF^*(\psi)\to HF^*(\psi\circ \psi)$$ which in the case $\phi=\psi={\rm id}$ reduces to the quantum cohomology ring of $M$ (Schwarz).
An interesting special case is where $M=M_\Sigma$ is the moduli space of flat connections on a principal SO(3)-bundle $P\to \Sigma$ (with $w_2\not=0$) over a compact oriented Riemann surface $\Sigma$. The mapping class group of $\Sigma$ acts on this space by symplectormorphisms $\phi_f\colon M_\Sigma\to M_\Sigma$. The diffeomorphism $f\colon \Sigma\to \Sigma$ also determines a mapping torus $Y_f$ and there is a similar product structure $$HF^*(Y_f)\otimes HF^*(Y_g)\to HF^*(Y_{fg})$$ defined in terms of ASD-instantons on a suitable 4-dimensional cobordism. Atiyah and Floer conjectured that there should be a natural isomorphism of the symplectic Floer homology group $HF^*(\phi_f)$ with the instanton Floer homology group of the 3-manifold $Y_f$. This was proved by the author in collaboration with S. Dostoglou. A natural extension of this proof shows that this isomorphism preserves the respective product strucures. This can be used to distinguish isotopy classes of symplectomorphisms $\phi_f$ and $\phi_g$ (Callaghan).
Another interesting case occurs when $P\to\Sigma$ is the trivial SU(2)-bundle. Then every 3-manifold $Y$ with boundary $\partial Y=\Sigma$ determines a Lagrangian submanifold $L_Y\subset M_\Sigma$. In this case the Atiyah-Floer conjecture says that the Floer cohomology group of a homology-e-sphere $Y_{01}=Y_0\cup_\Sigma Y_1$ (where $Y_0$ and $Y_1$ are handle bodies) should be naturally isomorphic to the symplectic Floer cohomology $HF^*(L_{Y_0},L_{Y_1})$. The proof requires a removable singularity theorem for ASD-instantons on a half-space which at the time of writing is still an open question. There is also a product structure $$HF^*(Y_{01})\otimes HF^*(Y_{12})\to HF^*(Y_{02})$$ which under these isomorphisms should agree with the corresponding products in the symplectic case.