Can I keep my current nationality, or do I have to give it up to become British?

From the UK's side, becoming a British citizen does not mean giving up your existing nationality. The UK permits dual (and multiple) citizenship.

However, whether you can keep your other nationality depends on the laws of that country. Some countries automatically withdraw citizenship when you acquire another, while others fully allow dual nationality. Check your home country's rules before you naturalise so you understand the consequences.