Bus
Most bus companies stop at the Erzurum Otogar (the main intercity bus station), which is about 1 km away from the main city centre. It is a walkable distance to town, or you can take the public bus to/from the Otogar.
Buses from Trabzon are available (4.5hrs and 30LT, 25 if you don’t look like a tourist and don’t speak much english).
