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 non-trivial 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 pi tons, 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 willfocus on recent numerical developments in overcoming the curse of dimensionality in many-body two-particle calculations: the intermediate representation and quantics tensor trains methods.