This depends on the country you're travelling to. Many serious illnesses are caused by contaminated water, so it's essential you check the purity of the water in your destination before drinking it. Travel guides should have a section on this.
Generally, European countries will have similar water purity to the UK - however, the culture is often completely different. In Germany, for example, restaurants can charge whatever they like for tap water. The German paper Focus even published a "dos and don'ts" warning tourists never to ask for tap water, as it feels that's akin to asking for free food in a restaurant!
For general advice on how to stay happy and healthy abroad, check out our 60+ Overseas Travel Tips or the NHS Choices website. For more in-depth information on the various health risks in each country, including an important warning about fake bottled water (always check seals are intact), see the National Travel Health Network & Centre (NATHNAC).