A design of a chiral ligand and the metal-coordinated catalyst is very important for achievement of a high optical purity in a catalytic asymmetric reaction. Recently, we developed the chiral N-P, S-P, and S-S ligands having N, O acetal and S, O acetal structures, such as phosphinooxazolidine (POZ), phosphinooxazinane (POZI), phosphinooxathiane (POT), and sulfideoxathiane (SOT). They afforded products in an excellent enantioselectivities in Pd-catalyzed asymmetric allylic alkylation, amination, and tandem allylation. Furthermore, novel catalytic asymmetric synthesis of ent-CP-99, 994 and ent-L-733, 061 were accomplished by using Pd-catalyzed allylic amination as a key reaction. Cationic Pd-POZ or Pd-POT catalysts demonstrated high levels of catalytic activity in the asymmetric Diels-Alder (DA) reactions of some dienes with oxazolidinone or pyrazolidinone typed dienophiles. Cationic Pd-POZ catalyst showed an excellent catalytic activity in the DA reaction in ionic liquid (IL). The catalyst could be reused eight times without significant decrease of yield and enantioselectivity in the use of IL/CH₂Cl₂ as a solvent. The review summarizes these our studies.