If you're hoping to stay connected to the outside world, and all of your friends and family, when travelling to Luxembourg then you have a couple of options. If being dependent on connecting to the local, free wifi isn't for you then you can activate your global roaming option in accordance with your mobile provider's international policies (you may incur an additional fee) or purchase a local SIM card.
Luxembourg uses the GSM network and while this shouldn't pose a problem for travellers from countries such as Australia, most of Europe, and the United Kingdom, it may pose a problem for those travelling from the United States with mobile phones attached to the CDMA network. Despite this issue, most smartphones will still connect to the local network upon arrival in Luxembourg, however, additional fees may apply so it's best to check with your network provider before you leave. If your phone is having issues connecting, buying a local SIM card is recommended.
Once you're connected one way or another, you should be able to use your phone in most urban areas of Luxembourg. Mobile phone coverage may decrease in more rural and mountainous regions of the country (such as in the Ardennes) and this will affect your ability to send and receive messages, as well as connect to the internet.