Yorkshire Water said that some areas were experiencing flash flooding due to the heavy rainfall.
West Yorkshire Police's customer contact centre said it was also experiencing a high number of calls about roads lifting as a result.
In Leeds, there were reports of tarmac lifting and floating in puddles in Town Street, Farsley.
There had also been reports of flooding in Lowood Lane, Birstall, and on the A650 in Wrenthorpe between The Wheel Inn and Ruskin Avenue.
Not every caller was ringing for the right reasons though, as the centre tweeted: "Ringing 999 for a lift home due to the heavy rain is NOT an appropriate use of this emergency facility. #timeforabrolly#notpolice"
Flooding at Baildon is caused disruption to rail journeys between Bradford Forster Square and Ilkley, but the line has now reopened.
National Rail said trains had been unable to run between the two stations until Network Rail had assessed the flooding.
West Yorkshire Metro said flooding in the Troydale area of Pudsey meant buses were currently unable to get through earlier today.
Services affected include First Bus's number 11 route but this has now been able to return to its usual route.
Meanwhile, North Yorkshire Fire Service said its crews had responded to reports of properties being flooded in Knaresborough, Boroughbridge, Dishforth and Richmond.
They have been using pumps to remove water from the affected homes and business, while also focusing on unblocking drains to reduce the risk of further flooding.