The works attributed
Apollodorus is credited by ancient sources with the Forum of Trajan and its basilica, the column, the bridge over the Danube built for the Dacian campaign, and works including baths at Rome. A surviving treatise on siege engines is transmitted under his name.
The Danube bridge is the most remarkable of these and does not survive. It is depicted on Trajan's Column and described by Cassius Dio, who regarded it as the most impressive of Trajan's works; the piers are reported to have been visible in the river long afterwards.
The quarrel with Hadrian
Cassius Dio tells a story in which Apollodorus dismissed a design of Hadrian's, was resented for it, and was eventually banished and put to death after criticising the emperor's temple of Venus and Roma.
The story is entertaining and is not well supported. Dio wrote a century later, the anecdote fits a familiar pattern of the artist punished for candour, and nothing corroborates it. This platform records it as a tradition rather than as a fact about how Apollodorus died.
Why the attributions matter
Roman architectural authorship is better attested than Greek, but not by much, and it is still mostly literary. Apollodorus is the most prominent named Roman architect after Vitruvius, and even in his case the connection to the buildings rests on later writers rather than on inscriptions from the works themselves.