Heavy rain and strong winds from Storm Babet have closed schools and roads in the districts.
Several roads were closed after trees came down, while the Wensleydale School in Leyburn shut early due to the risk of flooding in the building.
Leyburn Primary School also allowed parents to collect their children early to ensure everyone was able to get home.
The road between Redmire and Carperby was shut this morning after a large tree came down, although it was later cleared and the road is now passable.
Large puddles have been reported on several roads, although no rivers had flooded roads by 3.30pm this afternoon.
Firefighters from Richmond were called to a flooded property in Downholme, while a crew from Leyburn responded to reports of rising floodwater in the town’s Mount Drive this afternoon.
Crews used a light portable pump to divert water away from the properties.
An amber warning for rain and flooding is in place this evening for much of the district.
The warning is due to stay in place until 6am on Saturday.
The Environment Agency (EA) has advised of high levels on the Rivers Swale and Ouse. The EA is closing flood gates to help to protect York and Malton.
Council officials say partners in the North Yorkshire Local Resilience Forum (NYLRF), which brings together councils, emergency services and health organisations, have been responding to incidents and are prepared to support residents affected by the storm over the weekend.
North Yorkshire Police is reiterating the call for people to keep extra safe during the current damaging winds and heavy rain.
Officers are warning motorists to avoid making any non-essential journeys and to steer clear of flood water and other high-risk areas such as the coast, where waves are posing a danger to vehicles and pedestrians.
Superintendent Rachel Wood said: “Where temporary road closures have been put in place due to flooding or over-topping waves from the sea, it is essential to follow the diversions for the safety of everyone involved.
“If you find yourself stuck in flood water or in any other dangerous situations as a result of the storm, please make sure the emergency services are called by dialling 999.
“Please avoid using 999 for any non-emergency issues to ensure the emergency services can respond as quickly as possible to any serious incidents that may occur.”
Fallen trees of other debris in main rivers can be reported via the Environment Agency’s 24-hour number, 03708 506 506.
For debris on roads, contact North Yorkshire Council on 0300 131 2 131 or City of York on 01904 551550. Out of hours, call the police on 101.
Check road conditions before travelling using North Yorkshire Council’s road cameras at https://www.northyorks.gov.uk/cameras
Find tips on driving in rain or on wet and flooded roads at https://www.northyorks.gov.uk/winterdriving