I’m not talking here about Nordic or back-country skiing, but downhill skiing between two countries. In other words take a lift in the morning in one country, and have lunch in another.
There are a number of resorts where you can ski from one country to another (and back), but not surprisingly they are all in Europe.
Perhaps the most famous is the Matterhorn Ski Paradise which links Switzerland and Italy. Zermatt lies at the foot of the ski area on the Swiss side and Breuil Cervinia lies across the Italian border, with the majestic Matterhorn standing over both of them. Cervinia is cheap and cheerful, Zermatt not only provides the best views of the Matterhorn it is possibly the most complete ski resort in the world (and one of the more expensive).
The Silvretta Ski Arena bridges Switzerland & Austria, and there are even border control posts on the piste – although I have never seen them manned. Duty-free Samnaun lies on the Swiss side whilst the party town of Ischgl is in Austria.
Les Portes du Soleil is a huge sprawling resort between France & Switzerland, with o650km of piste. There are a whole bunch of ski resorts in the circuit, with Avoriaz and Morzine in France and Champéry in Switzerland amongst the more notable.
The Milky Way between France & Italy is not quite as big, but with 410km of piste is still one of the largest ski areas in the world. Montgenèvre lies in France, whilst across the border in Italy are Clavière, Cesana, Torinese, Sestrière, Pragelato, San Sicario and Sauze d’Oulx.
Espace San Bernardo links La Rosière in France with La Thuile in Italy.
Kanin-Bovec-Sella Nevea is one of the newer cross border resorts, linking Italy and Slovenia.
Nassfeld-Lake Pressegger is a little known resort in Carinthia, but it has 100km of piste and has runs that cross the Austrian border into Italy, and it is possible to have lunch on the Italian side of the border.
Not strictly speaking a cross-border resort but a section of Gstaad Mountain Rides links the Swiss German part of Switzerland with the French-speaking part, crossing the Röstigraben. So it is possible to take up a lift from Rougemont to the La Videmanette ski area from where you ski or snowboard down to Chalberhöni and Gstaad.
Equipment available right here on Tom Wohrman Sports.
With a wealth of ski resorts to choose from, I chose to visit 
There has been precious little snow since November, and with freezing levels rising above 2000m at times, resorts have struggled to keep a significant number of runs open. Snow machines have been judiciously deployed, meaning many upper runs with snow cannon are in good condition.Resort runs shielded by trees and some lower lying North-facing runs are also looking good, but even with cannon many South-facing runs are patchy, with exposed sections. Almost 50 Swiss resorts have not been able to open at all, and less than 40% of Swiss ski runs are open according to information published by the Swiss Tourist Board.
Unprepared runs are generally closed and off-piste skiing is all but non-existent. Despite the bravado of some resorts, I doubt if there is anywhere in Europe with decent skiing outside of prepared runs with snow cannon. Annoyingly this means a lot of useful cut-throughs and alternative routes are closed, funnelling skiers and snowboarders trying to get back to resorts to a limited number of routes (or even needing to take lifts down). There is also something dispiriting about vistas customarily snow-covered being brown and bare – just see how Verbier looks now:
as opposed to normally in the season: