NOVELTY - A block copolymer comprising at least one segment having an ion exchange group and at least one segment having no ion exchange group, is new.    USE - As polymer electrolyte that is useful polymer electrolyte membrane, polymer electrolyte composite membrane, catalyst composition, and polymer electrolyte fuel cell.    ADVANTAGE - The block copolymer exhibits superior performance in various properties such as water resistance and ion conductivity. The ion conductive membrane obtained from the block copolymer exhibits high power generation property.    DETAILED DESCRIPTION - A block copolymer comprising at least one segment having an ion exchange group and at least one segment having no ion exchange group, is new. The at least one of the segment having an ion exchange group is the segment of formulae -(Ar1-C(=O)-Ar2-O-(Ar3a)k-O)m- (I), -(Ar1-C(=O)-Ar2-O-(Ar3a-R-Ar3b)k-O)m- (II), and -(Ar1-C(=O)-Ar2-O-(Ar3a-R)k-Ar3b-O)m- (III). The at least one of the segment having no ion exchange group is the segment of formula -(Ar4-S(=O)2-Ar5-O-(Ar6-(X)r)s-Ar-7-O)n (IV). The segment has ion exchange group density of greater than or equal to 4 meq/g.    m and n=greater than or equal to 5;    k=1 or 2;    Ar1, Ar2, Ar3a and Ar3b=divalent aromatic groups optionally having optionally substituted 1-20C hydrocarbon group, optionally substituted 1-20C alkoxy, optionally substituted 1-20C acyl, optionally substituted 6-20C aryloxy or optionally substituted 7-20C aryl carbonyl;    R=O, 1-6C alkylene or fluoro-substituted 1-6C alkylene;    Ar4, Ar5, Ar6 and Ar7=divalent aromatic groups optionally having optionally substituted 1-20C hydrocarbon group, optionally substituted 1-20C alkoxy, optionally substituted 1-20C acyl, optionally substituted 6-20C aryloxy, optionally substituted 7-20C aryl carbonyl or fluoro;    r=0 or 1;    s=0-2;    X=direct bond, -O-, -CH2-, -C(CH3)2-, -C(CF3)2-, -S-, -C(C6H5)2-, -C(=O)-, cyclohexyl-diyl or 9H-fluoren-9-diyl.    Provided that at least one of Ar1 and Ar2 has an ion exchange group. The Ar3a and Ar3b optionally have an ion exchange group. INDEPENDENT CLAIMS are included for the following:    (1) a polymer electrolyte comprising the block copolymer as a main component;    (2) a polymer electrolyte membrane comprising the polymer electrolyte;    (3) a polymer electrolyte composite membrane comprising the polymer electrolyte and a porous substrate;    (4) a catalyst composition comprising the polymer electrolyte and a catalyst component; and    (5) a polymer electrolyte fuel cell having at least one membrane among the polymer electrolyte membrane, the polymer electrolyte composite membrane and the catalyst layer comprising the catalyst composition.