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#1: Using the phone abroad
No signal reception (i.e. the reception triangle is empty, or has a magnifying glass in it, or a cross):
Note that you need be in range of network that is compatible with your phone and for which your provider has a roaming agreement. To check the networks in the country where you are compatible with your phone, see for example willmyphonework.net. Normally negotiations with the network are automatic (so you don’t need to know which network your provider has an agreement with), but in some rare cases providers offer discounted rates when connected to specific networks.
That being said, in most cases things should work. So if they aren’t working: check whether network selection is set to automatic (Settings > [Wireless and Networks] More… > Mobile Networks > Network Operators). If it is, sometimes setting it to manual, attempting to connect to any network (even if it fails) and then back to auto helps. In some cases turning the phone off and on again also helps. Whilst you’re in the settings, it’s a good idea to check whether the data roaming settings are also the way you want them. Note that finding a network when abroad may take longer than you’re used to at home, at least on the first attempt (cases where it takes ~5 minutes aren’t unheard of).
There is a signal, but calls/SMS don’t work: If you do have connection to a network (i.e. there is a signal strength indicated), but calls/sms/data don’t work, then often it’s determined by something connected to the account (i.e. no valid credit for use abroad; a block on your account for use abroad). This can be something at your provider’s end, but there are some SIM settings to check as well. Find them via the phone app (the one you use to make phone calls): tap the menu key > Settings > Voice Call > Call Barring. Make sure the relevant Call Barrs are disabled.
Data doesn’t work. Double check Settings > [Wireless and Networks] More… > Mobile Networks for the roaming options (Under Settings > SIM management on the FP1). Check whether your calling plan includes data access. Check whether the network you’re connected to provides data services at a frequency that your phone can work with (see link under ‘no signal reception’). In rare cases manually falling back to 3G or 2G helps.
Some SMS are not received either by you or someone you sent them to. Check whether the number you are sending messages to has a country code set. More often, however, this occurs when there is no agreement on how SMS is handled between your home provider and the local network provider that you are connected to. In some cases this may even mean that messages are delivered in one direction only.
The same SMS is received over and over again As with the issue above, this has to do with how SMS is handled on different networks. The semi-technical explanation is that when a message gets delivered, the phone normally acknowledges that the message was received (this is different from the receipt notifications that can be sent to users). In some cases that acknowledgement never reaches the message centre that the message was sent through. The message centre will then assume the message was lost, and will attempt to resend the message at increasingly large time intervals until it either gets acknowledgement that it did arrive, or fixed time period runs out.
#2: Roaming when not abroad
Roaming needs to be activated to use data. This occurs on some specific operators that use the network of another operator. See this topic for more information.