The gardens of the former Royal Palace at Vrana, along Tsarigradsko Shosse in the direction of Plovdiv are open to the public at the weekend.
While not exactly inspiring, the gardens do provide a haven of peace and quiet away from the city traffic and since the entry fee of 5 leva is not exactly cheap by local standards there are relatively few visitors. Try to wait for a guided tour, which leaves on the hour every hour, and is free of charge. The tour guide will point out the more interesting plants (mainly trees) and tell you one or two stories about life at the palace. We were surprised to hear that two elephants had been kept here for farm work.
German and Austrian landscape architects were hired to create the palace grounds after Tsar Ferdinand bought the land 1889 and the park continued to develop over the next 40 odd years until the state repossessed it.
The palace fell into the hands of the state after Tsar Simeon was exiled in 1945 and was returned to him a few years ago. Simeon Saxe Coburg Gotha handed the grounds back to the municipality with the condition that it be opened to the public. He and his family still use the palace building itself.
To be honest, the landscaping could be better - it is apparently maintained by the company that maintains all of Sofia's parks - but there are some interesting and exotic tree specimens. It's also worth going to have a peek at the palace and its strange architecture.
To get there driving from Sofia you would have to drive down Tsarigradsko Shosse and turn right at the junction with the ringroad following the brown signs. You pay one lev for parking. Alternatively if you are travelling by public transport you can get off the number 505 bus (from Orlov Most) at the entrance.