Considering　some　helical　flows　of　a　generalized　Oldroyd-B　fluid　between　two　infinite　concentric　cylinders,　which　is　due　to　the　cylinders　oscillate　around　their　common　axis　and　accelerating　slide　in　the　direction　of　the　same　axis　with　prescribed　velocities,　this　paper　obtained　the　exact　solutions　of　some　unsteady　helical　flows　by　using　Laplace　transform　coupled　with　Hankel　transform　for　fractional　calculus.　The　results　show　that　the　smaller　the　values　of　α,　the　more　slowly　the　velocity　decays　with　the　flow,　and　the　material　parameter　β　has　quite　the　opposite　effect　to　that　of　α.　The　corresponding　solutions　for　generalized　second　grade　fluid,　generalized　Maxwell　fluid,　Newtonian　fluid　and　generalized　Oldroyd-B　fluid　through　a　circular　cylinder　were　obtained　as　limiting　cases　of　general　solutions.