Susceptibilities and optical conductivity are examples of two-particle response functions that are the key quantities for connecting theoretical predictions with experimental results. It can however become highly nontrivial to calculate them, especially in cases when vertex corrections are important. In my talk I will explain what vertex corrections to response functions are and when they become important. I will then present diagrammatic methods of their computation with the focus on parquet equations. I will show results for a special type of vertex corrections to optical conductivity, the ⇡ tons [1], that are present in systems with antiferromagnetic or charge density wave fluctuations. These vertex corrections lead to a displaced Drude paek in correlated metals. In the last part of my talk I will outline challenges faced when solving equations for two-particle quantities. I will focus on recent numerical developments in overcoming the curse of dimensionality in many-body two-particle calculations: the intermediate representation [3] and quantics tensor trains methods