Council workers began the process of removing trees which had fallen on to roads across the city, with the local authority's forestry teams said to be working "flat out".
The council said it was unable to quantify how much damage had been caused, but the bill is likely to be sizable.
Among those suffering were pupils at East Linton Primary School in East Lothian, who were forced to move classrooms after winds caused "considerable damage" to the rear of the school.
East Lothian Council said nursery classes would be closed on Thursday and Friday due to the damage.
Elsewhere, the Capital's bus station was closed due to storm damage, with services arriving and departing from St Andrew Square until repairs can be made.
Winds of up to 80mph tore through the city on Monday, bringing widespread disruption to transport, felling trees and causing masonry to fall.
Councillor Robert Aldridge, environmental leader, said: "We are working around the clock to clean up all the debris and damage from this latest bout of severe weather."