If everyfinitesystemofpolynomialequationsoveraring R has asolutioninthe ring R if andonlyifithasasolutionin ˆR where ˆR representsthecompletionof R, then wesaythatthering R has the Artinapproximationproperty. M.Artinsetin a numberofconjectures,thefollowingtheoremsolvedoneofthemwhichsays,“an excellentHenselianlocalringhasthepropertyofArtinapproximation”.General Neron Desingularizationisthebaseoftheproof. Let R and R0 beNoetherianrings,foraspecial(thatisregular)morphism u : R ! R0, any R-morphism '' : S ! R0 with afinitetype R-algebra S, factors through an R-algebra T whichissmooth R-algebra, thatis, '' is acomposite Rmorphism of S ! T and T ! R0. The R-algebra T is calledaGeneralN´eron Desingularization (shortlyGND). In ourthesiswegivetheconstructiveproofofGeneralNeronDesingularization for thecasewhen R and R0 are localringsofdimension m and S has abigsmooth locus,wealsogiveauniformGeneralNeronDesingularizationforlocalringsofdi- mension m along withthealgorithmstoconstructtheN´eronDesingularizationin these cases.Anothercontributionisthat,wegivethenestedstrongArtinapproxi- mation.