We investigate the conformal window of four-dimensional gauge theories with fermionic matter fields in multiple representations. Of particularly relevant examples are the ultra-violet complete models with fermions in two distinct representations considered in the context of composite Higgs and top partial-compositeness. We first discuss various analytical approaches to unveil the lower edge of the conformal window and their extension to the multiple matter representations. In particular, we argue that the scheme-independent series expansion for the anomalous dimension of a fermion bilinear at an infrared fixed point, γχ χ, IR , combined with the conjectured critical condition, γχ χ, IR = 1 or equivalently γχ χ, IR (2−γχ χ, IR ) = 1, can be used to determine the boundary of conformal phase transition on fully physical grounds. In illustrative cases of SU (2) and SU (3) theories with N R Dirac fermions in various representations, we assess our results by comparing to other analytical or lattice results. *