Resolutions above monitors native?

So, I’m just getting into this gaming thing and so far so good as I’m having loads of fun. Is there though, any benefit to running games as higher resolutions than your monitor? My current Dell runs at 1680×1050 native and it seems to me that running at the native would yield a better picture than if I turned up the resolution. Is there something to this? Should I limit the resolution to the monitors native? What do you guys do?