Abstract
We introduce the basic notions and the framework of the non-perturbative renormaliza tion group (NPRG) particularly for pedestrians using elementary examples. We derive an exact NPRG equation and set up its approximation method. The lowest order approxima tion called the local potential approximation (LPA) is applied to scalar theories with the ferromagnetic transition and quantum mechanics with tunneling. We compare our results with other methods, and will show that NPRG is very promising since it does not suffer any divergent series expansion. Then we analyze the dynamical chiral symmetry breaking in gauge theories and get the chiral phase structures. The LPA NPRG improves results by the ladder Schwinger-Dyson equation in a gauge independent way.
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