The real space decimation method has been successfully applied over the years to understand the critical phenomena as well as the nature of single particle excitations in periodic, quasiperiodic, fractal, and decorated lattices in one dimension and beyond. The power of the method is specially revealed through its application in lattice models, leading to an elegant understanding of the nature of the single-particle states and the corresponding transport properties. In this review, we discuss, using a variety of decorated lattices, how the realm of this method is extended to unravel diverse electronic phases of matter, such as the Dirac systems, or lattices exhibiting flat bands and topological phase transitions.