Rome defeated Carthage for the final time in 146 BCE. The tales of salting the fields are nonsense, and no ancient source supports this. The Romans did, however, unleash their fury on the city and made a pretty big mess of the place. Polybius tells us that Scipio thought of the destruction of Troy as he watched Carthage burn (Polyb. 38.22). The urban center of northern Tunisia became the nearby port city of Utica. The site of Carthage continued to be sporadically occupied in the years following 146, by refugees or looters or whatnot, but was no longer an important urban center. In 122 BCE, the Roman statesman Gaius Gracchus attempted to found a citizen colony atop the site of Carthage called Colonia Iunonia, but this effort was aborted by the Senate for political reasons. Seven decades later, Julius Caesar founded a veteran colony on the site (technically it was after his death, and the major work happened during the reign of Octavian).

This new city was Colonia Concordia Iulia Karthago. The ruins were cleared and the Romans undertook major building projects there right from the beginning. Old Carthage had a hill, called the Byrsa, upon which her most important temples sat. The Romans leveled the top of the hill when they founded the new colony, and 100,000 cubic meters of debris and dirt was pushed onto the flanks, creating a flat plateau on top. There a new forum and Roman temple complex was constructed. The flanks of the newly-shaped hill were held in place by massive Roman retaining walls. There has been some excavation in the spaces between these buried sections which has revealed "Hannibal's Quarter," a section of the Punic city's remains. I can hunt down some references for that, if people are interested (a lot of it is German language). We know virtually nothing about the Roman structures on top of the hill because there is no opportunity for excavation due to modern buildings.

A hundred years later, the new Carthage had become one of the largest and most important cities in the Mediterranean. Its port facilities were repaired and expanded, it had a circus and amphitheater, and opulent bath complexes. It remained an important city until late antiquity and the coming of the Islamic peoples, who founded Tunis nearby. Slowly Carthage was abandoned and forgotten. There has been a great deal of excavation in the modern period, usefully summarized by Serge Lancel in his book "Carthage" (helpfully translated to English!).