Right now I'm playing Shadowrun4, I know that there don't exist classes which restrict your character developement. Are there any other P&P RPGs out there, which also don't have classes?

That means particularly: I don't want ANY classes at all. The RPG should be for up to 5 players (plus GM). I don't want something like a tabletop: A plain P&P RPG where nothing else but your character stats and probably a set of dices is needed.

The genre is not important. Sci-Fi, Fantasy, anything will do. I should probably say that I'm also interested in "bigger" RPGs with a good and well-elaborated background story.

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    \$\begingroup\$ The answer to this is trivially "yes," and any more in-depth answer would turn into an unending list of literally hundreds of different systems. If you're looking for a game recommendation, we'll need more specific criteria to work with. How many players and what session/campaign length; what genre or tone; do you want miniatures on a grid or theatre of the mind; what features of character development are you looking for; and so forth. Tell us what you do want as well as what you don't, so we can make useful suggestions. If you'd just like to chat about the topic, come join the Role-playing Games Chat. \$\endgroup\$ – BESW Jul 28 '14 at 10:59
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    \$\begingroup\$ @Ferox Elaborating on lisardggY's comment, here are a few question that could help you tell us what you're looking for: Do you need one of the players to be the game master or are masterless games ok? Do you want it to focus on combat, character development (this usually does not work well with "deep, pre-written backstories", since players tend to cristallyze on those), creating a story togheter? Must it be a collaborative or a free-for-all game? Should it be able to manage long campaigns or just one-shots is ok? Setting/tech limitations might also be useful. \$\endgroup\$ – Zachiel Jul 28 '14 at 11:47
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    \$\begingroup\$ Think about your actual problem and then explain it to us. Are you looking for a game like this to run (in which case I bet you/your players have a lot more specific requirements than "whatever")? Or are you wanting to understand how character progression works without classes from a game design standpoint? \$\endgroup\$ – mxyzplk - SE stop being evil Jul 28 '14 at 11:53
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    \$\begingroup\$ In short, we can't tell you what you want; you have to decide that for yourself. When you decide what you want, we can help you find the perfect system to satisfy those wants. But we can't just throw systems at you until something sticks. \$\endgroup\$ – KRyan Jul 28 '14 at 15:21
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    \$\begingroup\$ Do note that something like 90% of RPGs - most of them except for the D&D's and other stuff made by TSR/WotC and other really old games (Palladium) don't have classes. Classes are actually unusual in game design but since D&D is what so many have played it seems like it's the default. \$\endgroup\$ – mxyzplk - SE stop being evil Jul 28 '14 at 22:49