Liptov is the best region if you are looking for downhill skiing or freeride skiing in Slovakia. The biggest ski centre Jasná is only 15km from the town centre. The best freeride skicentre is Čertovica (a bit to the East). On the map you can find all ski centres of the area. Crosscountry skiing is also a possible activity. Unfortunately there wasn’t much snow in the main valley for many years so if there is enough snow, it is possible mostly everywhere outside the town. On the map you can find some traditional crosscountry tracks with some descriptions.
One of the sports that is getting more and more popular every year in this area is ski touring. You can also find your way up to the mountain even easier than in summer when you have to follow the path strictly in many places. The best areas for backcountry skiing is the Žiarska Valley in the West Tatras and area of Chopok and Ďumbier in the Low Tatras. Also The High Tatras can give you many opportunities for skiing. Before you go for any outdoor activities do not forget the mountain insurance and to check the risk of avalaches together with the weather forecast for the mountains!