In the basement, the Hungarian National Museum has a very large "lapidarium", i.e., a collection of Roman inscriptions and reliefs. The museum also has a great collection of finds belonging to the Romans, Gepids, Huns, and Avars.
There are 20 items in Budapest, Hungarian National Museum:
Avar collier
Cretan larnax
Byzantine glass drinking horns from an Avar tomb
Byzantine jewelry from an Avar tomb
Dunaújváros, Relief of Gaius Mucius Scaevola
Tapioszentmarton, Scythian shield emblem in the shape of a stag