If a person sends threatening or abusive messages/phone calls, they could be committing an offence. The most relevant offences are 'harassment' and 'malicious communications'. See Q497 for further information regarding these offences.
The first thing to do is report the matter to your telephone company, regardless of whether it is a landline or mobile network. If necessary, they will put a trace on your line and instruct you to keep a diary of the calls, the time of the calls and also the type, for example, silent or heavy breathing etc. This will enable the telephone company to gather information and contact you again when they feel they have sufficient evidence.
The police need details of the offending number and evidence of calls made to pursue a prosecution. Once this is obtained, you may be required to attend your local police station to complete a form that will enable the police to have access to your telephone records.
This process can take some time and it is not unusual for weeks to pass before the police re-contact you about the complaint. In this time they will be aiming to identify the offender and where possible to make a prosecution. The process is the same if you think you know who the offender is; the police still require the evidence to prosecute.
If the calls/text messages are of a threatening nature and there is a serious threat to your safety, you should report this to your local policing team who will treat the matter as a higher priority.