Aerodromes are not closed due to high wind; the decision to operate rests with the operator/ pilot in command. The maximum value of wind (or crosswind) is determined and promulgated by each operator, taking into account the manufacturers recommended/maximum limits. Usually, in extreme high wind conditions, it is not the windspeed per-se that is the limiting factor, but the effect of any crosswind, turbulence, or windshear, any or all of which can be present to some degree in high wind operations. Ground handling is an issue, also. No point in landing somewhere if the ground staff are unable to service the aircraft or operate the airbridge etc. Typically, in my neck of the woods, the airlines start to call it quits if the wind is consistently above about 50knots, or the crosswind frequently exceeds 35. Have seen Boeings land in about 60. Not that common, though.
cokie - I assume your question refers to windspeed along the runway rather than across. I have landed 737s with gusting 65kts and the major problem is taxying when you turn across the wind especially if the surface is at all slippery. That, as 'Intruder' says, tends to put a top end on it. Strong winds can also cause havoc on the parking stands. I was turned quite a few degrees on stand by a strong wind and I have seen a badly chocked BA jumbo turned 45 degrees.
As 'SR' says, until you try to turn out of the wind or open anything, actual wind speed is not really a problem for a jetliner as they are designed to fly at high (relative) wind speeds.
Again, there is no general headwind limit. There may be a headwind limit for autoland (25 knots in the 747), but no general limit. In practical terms, as has already been pointed out, turbulence and other weather (rain, downdrafts, etc) will make it inadvisable to land when the surface wind is high, and may restrict ground operations before flight ops are restricted. I have landed in 50+ knot wind, but it is not a common practice for most commercial operators.