The typical case is that twin-engine airplanes can absolutely fly with one engine shut down. This being possible means that the extra engine provides redundancy; if such an airplane couldn't fly in that condition, the second engine would double the chance of a flight-ending engine failure occurring.
In order to keep the airplane controllable when one engine is at full power and the other is shut down, designers take steps like keeping the engines fairly close to the fuselage so that the "leverage" -- the moment arm -- of the asymmetric thrust is reduced. They also ensure that the tail surface and rudder -- which provides the force to counteract the yawing moment caused by the unbalanced forces -- is large enough to do so under any expected conditions.
From a pilot's perspective, the greater the airspeed, the more control authority the rudder provides, so in an engine-out case, you want to keep the airspeed high enough that your rudder input can balance out the asymmetric thrust. It's also possible to add a little bank toward the operating engine, which also helps to balance things so the airplane flies straight ahead rather than turning continuously toward the failed engine.
For a 4-engine aircraft that's lost 3 engines, the additional concern is a lack of power to keep the airplane aloft. Depending on the weight of the aircraft, a single engine might keep the airplane in level flight without losing airspeed, but more probably some amount of descent would be required to avoid losing speed (and, eventually, once the speed decayed enough, losing control as well). How far you can get would depend on all sorts of factors, such as how heavy you were to start with, how much altitude you have to work with, if the winds are pushing you along, how much other damage is adding drag to the aircraft, and so on. Maybe the B-17 can't get all the way back to England with 3 of 4 engines shot out, but if they can get to France, or to the English Channel, the odds of avoiding capture after bailing out get better.
If you had a light enough 747 or A380, you might maintain level flight with one engine, but at most weights you'd probably be trading altitude for airspeed. With two of four engines operating, you could hold level flight at some significantly higher weight, though probably well below max takeoff weight. Any 4-engine aircraft should be able to fly fine on 3 engines, since otherwise you'd be at the point where any (single) engine failure on takeoff is unsurvivable, and that's not an acceptable scenario.
Altitude is also a factor; losing an engine up high near the service ceiling may mean a "drift down" to an altitude at which max continuous power on the remaining engine(s) will hold level flight + maintain airspeed. If that altitude is above the terrain, then all is good; if it isn't then the crew will need to use the time & distance available while drifting down to head toward lower terrain, or (worst case) toward wherever they prefer for the forced landing to take place.
The max load you can carry might have to be reduced so that you aren't too heavy to keep flying with one engine out; pilots regularly practice (in the simulator) what happens if an engine fails right at takeoff, just after the point you can no longer stop the aircraft on the runway. If you (and the aircraft) can fly out of that scenario, all other cases are less demanding. How much weight you can have at takeoff, based on engine-out climb performance (and other criteria) is something that gets computed based on the particular conditions of "this" flight.
Left to its own -- i.e. if the pilot didn't make any control inputs to counter the asymmetric thrust, then the aircraft would continuously turn into the failed engine, and things wouldn't end well. But, part of the training required to be able to fly a multi-engine aircraft is learning how to do so even with an engine failure. That's been the case basically since aircraft with more than one engine started flying, and certainly continues today with 2- and 4-engine airliners.