It’s normally a matter of courtesy for employees to work out their notice period. Although an employer can potentially pursue a breach of contract claim in cases where a contractual notice period has not been honoured, in practice this is very rare and usually is not worth the expense of legal action.
Employers may decide to obtain an injunction to prevent employees working for a direct competitor during their notice period, or sue for additional costs - or even loss of business - arising from a breach of contract, but these claims are usually only brought against senior executives.
Employees should bear in mind that they may need to ask their employer for a reference unless one has already been provided, so not completing their notice period may result in a less favourable testimonial (although any information contained in a reference should be accurate).
Some employment contracts may allow for a deduction of the final pay if an employee leaves before their notice period is up (see below).