In the first one a Taiwan Golden 202 Classic motorcycle was stolen from a garden at a house in Eden Place in the town. It is valued at £180 and has not yet been recovered.
In a second theft, a Honda 125 scooter was stolen from a house in Craignair Park in the town. This scooter was later recovered a short distance away in Stapleton Road.
Constable Victoria Urwin at Annan said: “We are not sure about the motive for these thefts and it may well be that joyriding is why they have been taken.
“I would urge anyone who may have any information about who is behind these crimes to call us at Annan on the 101 number.”
She added: “I would also ask scooter and motorcycle owners to be extra careful at this time about the security of their machines.”