Before cultural contact with Greece, Roman culture had forms of entertainment including athletic and gladiatorial contests, musical performances, perhaps mime, and other acts that share much in common with modern circuses such as jugglers. Roman drama per se was built mainly on Greek models.
The comedies of Plautus and Terence were usually translations or adaptations of Greek New Comedy, especially of the works of Menander. They usually had romantic plots and many stock characters including clever servants, lustful old men, innocent young lovers, boastful soldiers, and evil pimps. While actors were all men during the Republic, during the Empire, women began to appear on stage.
The main Roman tragedies that have been preserved are those of Seneca, which are also based on Greek myths, but are not direct adaptation of Greek playwrights. They tend to be more melodramatic than Greek plays.
Dramatic adaptations of the Gospel story began in medieval England. They slowly evolved from minor variations of the Latin liturgy (especially Easter) into vernacular dramas retelling key parts of the Gospel for those who were illiterate and did not know Latin. As this happened some five centuries after the fall of the Roman Empire, it is usually not considered part of Latin dramatic literature but rather part of the evolution of drama from classical to early modern forms.