The South Wales Fire and Rescue Service reported that emergency services have been carrying out rescues, evacuations, and welfare checks since the incident was declared in the town at 01:30 GMT on Saturday.
A severe cold snap is expected in Britain next week, with rain expected to continue in many areas on Saturday.
Nearly 50 flood warnings have been issued for England, while nine flood warnings have been issued in Wales.
Many businesses and homes in Monmouth town center were submerged as the Monmouth River rose. Some locals were evacuated to the local library amid warnings of a "life-threatening situation."
Local County Councillor Martin Newell, who owns a business in the town, said, "I haven't seen such bad conditions in Monmouth for 20-30 years." He added that he felt "deeply saddened" for those affected.
A family described the moment they were left "holding on to a tree" after floodwaters submerged their car while traveling from Hereford railway station to Abergavenny.
Monmouthshire County Council has urged drivers to avoid traveling in the local area.
The South Wales Fire and Rescue Service said teams have been called to 82 flood-related incidents across the wider region.
Matthew Lehnert, chief meteorologist at the Met Office, said up to a month's worth of rain is expected in some areas between Friday and Saturday.
Flood protection measures have been implemented in Bewdley, West Midlands, and emergency services in Shrewsbury will do the same on Saturday.
The British Red Cross said it has placed its emergency response teams on standby to help communities most at risk.
Friday's torrential rain has caused some travel disruption, and National Rail has warned of delays and cancellations over the weekend.
The AA has advised against traveling in "hazardous weather," saying: "Safety is paramount. Conditions can change rapidly, so stay updated and ensure you are prepared before traveling."
Some events have already been disrupted by the weather, including the Christmas event at Macclesfield – Lapland UK – which had to be canceled due to a fallen tree.
Organizers have said they will inspect the track at Cheltenham's Prestbury Park on Saturday morning ahead of the scheduled seven-race Paddy Power Gold Cup card to determine whether racing will continue.
Storm Claudia – named by the Spanish Meteorological Agency (AEMET) – has already brought heavy rain and strong winds to Spain and Portugal.
The storm comes at a time when the Environment Agency has warned that England will face widespread drought next year due to a rainless winter.
The country received only 83% of its average rainfall from January to October, and has suffered the driest spring in 132 years and the hottest summer on record.
It said the situation remained "dangerous" despite recent rains.