Generally, in common law countries civil cases are now tried by judge alone (a bench trial). However, some civil cases are still tried by jury, and in those cases the general rule is that the jury determines the quantum of damages, subject to the usual rights of appeal from discretionary decisions. Here are some references for four major common law jurisdictions, from most to least jury-friendly.
In the United States, in civil cases, the ‘right of trial by jury [is] preserved’ by the Seventh Amendment. While the constitutional right can be waived, and is not binding on the States, juries still commonly decide civil trials. In doing so, they assess the quantum of damages: see for example Johnson & Johnson Told to Pay $4.7 Billion in Baby Powder Lawsuit (Tiffany Hsu, New York Times, 12 July 2018).
In Canada, ‘depending on the suit and the court, the defendant may have a right to a trial by judge and jury’: Civil and criminal cases (Department of Justice). Nonetheless, civil juries seem common in Canada and determine both liability and the quantum of damages: Scrapping the civil jury (Michael McKiernan, Canadian Lawyer, 25 September 2017).
In the United Kingdom, civil cases are usually tried by judge alone. In John v MGN Ltd  QB 586, the Court of Appeal noted that ‘personal injuries [have] ceased to be the subject of trial by jury and bec[o]me in practice the exclusive preserve of judges,’ while jury trial had ‘surviv[ed]’ in defamation actions. When the jury determines liability, it will also assess damages: Luke Cooper's case shows damage of abolishing trial by jury in libel cases (The Guardian, 29 June 2012).
In Australia, trial by judge alone is the general rule, but in defamation cases the jury may determine liability while the trial judge assesses damages: Rebel Wilson awarded $4.5m in damages over defamatory magazine articles (ABC News, 14 September 2017). This follows from s 22(3) of the (largely) uniform Defamation Acts 2005 (eg. NSW, Victoria). Even by 1991, Australia had diverged from other common law countries – the High Court noted that ‘in all Supreme Courts other than that of New South Wales, a judge assesses damages for defamation in the absence of some contrary order or election’: Coyne v Citizen Finance Ltd (1991) 172 CLR 211.