Back in 2000, Sega released a stylish, original game called Jet Set Radio, or Jet Grind Radio at other territories. The cel-shaded graphical style was great, even on the Dreamcast at the time. Sega has been a roll on bringing back arcade, Genesis, and Dreamcast games to the HD era this console generation. It was only a matter of time Jet Set Radio gets the HD treatment on the Playstation 3, Xbox 360, and PC. While the visuals and soundtrack as still as special as they were back in 2000, the gameplay simply has not aged well with today’s standards. 6.0/10
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The Xbox sequel (or reimagining) Jet Set Radio Future , makes it up to you in every way possible. Despite making your player much faster, it gives you no time limit and the only time that police chase you is as an easy afterthought right at the end of a level. It has more platforming and rail grinding goes up telephone wires as well. It's a great, leisurely, game dressed up as an action packed one and the music is slightly better than in the original game in my opinion.
However the original game has a cel shaded style that is so pure that the sequel offers a different, more shaded and effects heavy, version of it, not necessarily a much better one.
In short: play the first game if you want a hardcore cat and mouse chase between you and the police (where you're the mouse).
Play the Xbox game if you love easier platform games.