Bronsted acid catalyzed (3(+) + 2) cycloadditions between hydrazones and alkenes provide a general approach to pyrazolidines. The acidity of the Bronsted acid is crucial for the catalytic efficiency the less acidic phosphoric acids are ineffective, while highly acidic chiral N-triflylphosphoramides are very efficient and can promote highly enantioselective cycloadditions. The mechanism and origins of catalytic efficiencies and selectivities of these reactions have been explored with density functional theory (M06-2X) calculations. Protonation of hydrazones by N-triflylphosphoramide produces hydrazonium-phosphoramide anion complexes. These ion-pair complexes are very reactive in (3(+) + 2) cycloadditions with alkenes, producing pyrazolidine products. Alternative 1,3-dipolar (3 + 2) cycloadditions with the analogous azomethine imines are much less favorable due to the endergonic isomerization of hydrazone to azomethine imine. With N-triflylphosphoramide catalyst, only a small distortion of the ion-pair complex is required to achieve its geometry in the (3(+) + 2) cycloaddition transition state. In contrast, the weak phosphoric acid does not protonate the hydrazone, and only a hydrogen-bonded complex is formed. Larger distortion energy is required for the hydrogen-bonded complex to achieve the "ion-pair" geometry in the cycloaddition transition state, and a significant barrier is found. On the basis of this mechanism, we have explained the origins of enantioselectivities when a chiral N-triflylphosphoramide catalyst is employed. We also report the experimental studies that extend the substrate scope of alkenes to ethyl vinyl ethers and thioethers.