You could have a bad coil/magnetto. Or it just might need to be cleaned (sanded to get the rust off of it and the flywheel.
The fuel filter is probably clogged and restricting the fuel flow. If the engine blows black smoke before it cuts out, then the engine is flooding - look for a dirty air filter or a fouled spark plug.
The fuel mixture is probably set to rich. Also, the wrong spark plug (with regard to heat range) could be in the engine. If you have an adjustable carburetor with mixture adjustment screws, try turning the screw (probably located on the bottom of the carburetor, in the middle of the bowl) inward 1/8 to 1/4 of a turn. This will lean the mixture. If you have a no-adjust carburetor, then there is not a lot you can do to change the mixture. Try using/replacing the spark plug with the correct spark plug as recommended by B&S. Good luck!
flywheel key could be sheared OR check valve settings as likely the cam followers are on the wrong valve.
The most comon cause of an engine that keeps revving up and down (aka surging) is a dirty air filter.
You have your lines on wrong. Make sure you have the one with the arrow in coming from the tank, the one pointing out goes to the carb and the one with no arrow goes to the last one
remove spark plug, remove air cleaner and hold hand over top of carburettor and pull start cable a few times , then reassemble and restart, also check fuel quality, ie water contamination
check to see if it is getting spark . you probably have the "kill mechanism " installed wrong . It's probably grounding out the ignition system. The "kill mechanism" is the cable operated mechanism that goes from the handle with a cable to the pull start housing . Its either improperly installed or out of adjustment.
No compression????
Rust on the fly wheel and or coil.
possibily its the wrong plugs could be hitting the pistons
it is losing its prime... i have the same problem right now and i have cleaned the carburetor, changed the gas and spark plug and changed the air filter. The only thing that i can think of is that there is something wrong with the priming bulb or the gas cap is hindering the air flow... still trying to repair....
wrong spark plug wire configuration. Check to see if two are backwards.