Of course yes .
Crosswind landings , like the one in the video you linked, are very common.
In fact, landings with no wind or only headwind are rare.
There are
several techniques pilots are taught during their extensive training to land aircraft when there is crosswind.
The Wikipedia article I referenced above lists them.
Since your question isn't about them, we won't discuss them.
Certainly, extreme weather situation are not very common.
Some parts of the world experience extreme weather more than other areas.
Diverting to other airports is not always an option and in some cases, once an airplane is on the approach, going around becomes impossible .
aviation.stackexchange.com/users/65/Farhan
In the US it is required by law to be trained in cross wind landings.
For large aircraft that require a type rating... §61.31 Type rating requirements, additional training, and authorization requirements... ...
(2) Received a logbook endorsement from an authorized instructor who has found the person proficient in the operation of the aircraft and its systems.
(i)
Normal and crosswind takeoffs and landings ; (ii) Wheel landings (unless the manufacturer has recommended against such landings); and (iii) Go-around procedures.
On a side note cross wind landings are very important since not all airports have 2 runway options.
Some large commercial airports will have 2 crossing runways which will allow you to pick the best one for the days winds (they are usually built to the general prevailing winds in the area).
This does not mean there will be no cross wind
but it does allow you to mitigate it.
Smaller airports may only have a single runway (or 2 parallel runways) which means you have no method of mitigation for the crosswind.
There are even some big airports that are only single runway due to space or geographic restrictions.
It should also be noted that airplanes them selves have a demonstrated cross wind component to which they are tested to be safe to land in a cross wind of said velocity.
This is not necessary a maximum (although it can be) but it is often used as such.
As for how often this happens that is largely a factor of the airport and the local weather.
For example I used to fly out of KPNE
they have 2 runway options (6-24 and 15-33)
they would use what ever was the best runway for the wind that day unless you needed the precision approach on 24.
This made landing there easy (150 x 5000 or 7000 helped as well) as there was never really more than a 15 degree crosswind.
I now fly out of KDYL which is a single runway airstrip that almost always has a crosswind, really keeps you on your toes.
Cross wind approaches are perfectly safe and are not out of the ordinary.
I have seen some people mistake the phrase Uncoordinated Flight as meaning a sideslipped crosswind landing is unsafe which is incorrect.
The plane can be uncoordinated because it is side slipping to the landing which is done deliberately by the pilot.
aviation.stackexchange.com/users/3677/Dave
The answers so far deal with commercial flying, from a recreational perspective pilots are only as well trained as the amount they practice.
Commercial organizations generally train their pilots for crosswind landings in simulators, and they usually fly enough to get real crosswinds often enough to keep in practice.
Private pilots are trained in crosswind techniques in initial training, keeping good crosswind abilities is up to the pilot once they pass their test.
I personally practice crosswind take-offs and landings as often as I can, however I know many pilots who won't fly if there's more than 5 knots across the runway.
aviation.stackexchange.com/users/1791/GdD
Yes, pilots are trained for crosswind landings.
I trained on a Cessna 172 as Paso Robles.
The video was completely normal.
A roll of < 10 degrees is not serious at all, well within a competent pilot's operational abilities.
The news story was written by someone who doesn't fly, thought it looked 'scary' and filled 90 seconds with nonsense, IMO.
aviation.stackexchange.com/users/9551/J Presper Eckert
I'm a little late to the party, but here is my take.
In that landing I see turbulence
but I don't see an obvious crosswind component.
In any case, dealing with crosswinds is something that happens more often than not when landing (at least at many airports and airfields).
So every single pilot has dealt with this since the very first days of basic training.
As for the wing rocking, correcting an off-level wing is probably the very first thing a student learns when getting acquainted with the aircraft's controls.
It is completely intuitive for every single pilot.
aviation.stackexchange.com/users/690/Martin Argerami
A  YouTube footage  of a KLM Boeing 777 landing at Schiphol airport in the Netherlands is certainly scary as the plane bumps and weaves and rolls from side to side.    Are pilots generally trained to handle such eventualities (or are such landings common in certain regions of the world?) or was everyone involved including the pilot taking a risk in this case without diverting the aircraft to a nearby safe location?
Are pilots generally trained for crosswind landings?
aviation.stackexchange.com
<landing><flight-training><crosswind>
aviation.stackexchange.com/questions/17302
aviation.stackexchange.com/users/6833/Chethan S.
Of course commercial pilots are trained in crosswind landings. In fact in the US this is a legal requirement. In many parts of the world this is a regular climatic condition at airports. Simulator training is also given and working as a pilot, they experience these landings on a regular basis. Yes, pilots are trained for crosswind landings. This is a legal requirement in the US. 

Crosswind landings are safe and they are performed frequently, particularly in parts of the world where extreme weather is more common. Diverting to a nearby location is not always possible.

Commercial pilots tend to get plenty of practice with crosswind landings, whereas private pilots may have to work harder on their ability to land in crosswinds.
Yes, pilots are trained for crosswind landings. This is a legal requirement in the US. Commercial pilots tend to get plenty of practice with crosswind landings, whereas private pilots may have to work harder on their ability to land in crosswinds. Extreme weather is more common in certain parts of the world. Diverting to a nearby location may not always be possible. Crosswind landings are common and safe.