  We employ the functional renormalization group approach formulated on the
Schwinger-Keldysh contour to calculate real-time correlation functions in
scalar field theories. We provide a detailed description of the formalism,
discuss suitable truncation schemes for real-time calculations as well as the
numerical procedure to self-consistently solve the flow equations for the
spectral function. Subsequently, we discuss the relations to other perturbative
and non-perturbative approaches to calculate spectral functions, and present a
detailed comparison and benchmark in $d=0+1$ dimensions.
