Provided are: a polyacrylonitrile-based precursor fiber for the production of a carbon fiber having a large single-fiber fineness, said precursor fiber ensuring high heat stability of a spinning dope and excellent productivity; and a copolymer suitable for the production of said precursor fiber. Also provided are: high-quality carbon fiber bundles which have a large single-fiber fineness and excellent productivity; a process for producing the same; and a process for producing flameproofed fiber bundles suitable for the production of the carbon fiber bundles. A polyacrylonitrile-based copolymer which comprises 93.0 to 99.4 mol % of acrylonitrile units, 0.5 to 4.0 mol % of (meth)acrylamide-based units, and 0.1 to 3.0 mol % of unsaturated carboxylic acid hydroxyalkyl ester units and in which the (meth)acrylamide-based units are (meth)acrylamide units and/or (meth)acrylamide derivative units having a molecular weight of 105 or less; a precursor fiber comprising the copolymer; a process for producing flameproofed fiber bundles which uses precursor fiber bundles consisting of the precursor fiber; carbon fiber bundles; and a process for production thereof.