Not many goody two-shoes back in the days of ancient Rome, especially before the coming of Christianity. Here are, in my opinion, the societal crimes of the Roman Empire listed roughly in order of magnitude:
(Tied for first place) Child sex abuse; slavery
(Tied for second place) Subjugating people of other nations; cruel and unusual execution methods
Deriving pleasure from violent spectacles
Environmental damage and extirpation of species
First, an important note. We have to be cautious about holding ancient societies to modern standards. Context is important. Many of the practices here were part of other cultures of the time, not just Roman. One example is slavery. Although slavery is never acceptable, it was part of human society for thousands of years before the modern era. Slavery wasn't practiced only by ancient Romans, but by many other European, African, and Middle Eastern societies of the time. So as you read, keep that in mind. Child sex abuse is another issue that may have existed for centuries before the first glimmers of modernity. Broadly speaking, it wasn't until the 19th century that Western Civilization began to pass age of consent laws for sexual activity.
Child Sex Abuse and Infanticide
Underaged children could be used for sexual pleasure of any kind, particularly if they were slaves. In most circles this was not considered wrong. A possible exception might have been if the child in question was from a noble family.
No age of consent. Such a concept didn't even exist in many ancient societies, although the marriage of a young woman was usually delayed until the bride-to-be had started menstruating.
Prior to Christianity, and in the absence of reliable birth control, an unwanted baby was left exposed at birth, where it would either perish or be picked up and taken in by other adults (see slavery below).
Roman Slave Market, Le Marché aux Esclaves, by Gustave Boulanger, 1886; via Wikimedia Commons and in the public domain.
Slavery
Even the Christian Church didn't ban slavery. Slave owners were merely exhorted by bishops to treat their slaves well.
Slavery in the Roman Empire wasn't based on skin color, like it was in the American South. In Roman times a favorite source of slaves was defeated countries.
People could become slaves by losing a battle to Roman troops, by piracy or kidnapping, or being born to other slaves in the household.
A slave was not considered to have legal personhood but akin to domestic livestock, to be used however the owner saw fit. They could be and often were abused physically, emotionally, and sexually.
Exposed infants could be claimed and raised by slave traders, who would eventually sell them into slavery.
Occasionally, adults in debt sold themselves into slavery as bond for loan repayment.
Sometimes desperate parents had to sell their children into slavery when they could no longer feed or care for them.
Invading and Subjugating Other Nations
Nations could either accept Roman rule or be destroyed by the military. Groups that quietly accepted Roman rule were generally relatively well treated...until tax time, that is.
The Roman state plundered whatever they could glean from invaded countries, usually in the way of slaves or natural resources.
Romans rationalized that they were improving non-Roman societies by bringing civilization, infrastructure, laws, and culture. By the mid-4th century AD, that extended to bringing forced conversion to Christianity.
If conquered peoples gave them too much trouble the Romans didn't hesitate to wipe them out. For example, after a Jewish revolt that began in AD 66, the Jews in Palestine were defeated, the temple plundered and destroyed, and the Jewish population denied access to Jerusalem, triggering the Jewish diaspora.
Cruel and Unusual Deaths
Although most people associate Rome with crucifixion, Roman used many other methods of execution, most of them equally unpleasant. (Note: The Romans didn't invent crucifixion. It was practiced by other civilizations that predated the Romans.)
Roman nobility were usually spared the cruelest types of executions and were given the choice of killing themselves (prior to Christianity, that is) or being executed quickly by soldiers. Unfortunately those of the plebian or slave class could not hope for such mercy.
Romans considered it amusing to dress up condemned criminals as mythological figures and make them act out dramas from mythology. For example, a condemned criminal might be made to put on wings like Icarus, then pushed from a high place so he could plummet to earth like the mythological figure. This was for the entertainment of Coliseum attendees (see below).
Executions could also be by burning alive, beheading, being shot with arrows, or strangling.
Damnatio ad bestias meant being sentenced to die via attacks from wild animals in the Coliseum.
Patricides were drowned by putting them into a sack filled with snakes.
Leopard attacking a criminal in the Coliseum, Roman floor mosaic, 3rd century AD, Archaeological Museum of Tunisia; via Wikimedia Commons and in the public domain.
Violent Spectacles
For centuries, bloody spectacles were one of the favorite forms of entertainment in Rome. During special occasions, these events could last for days, weeks, or even months.
Thanks to the Christians, gladiatorial combat ceased in the early 5th century AD.
By the 5th century AD, the most egregious Coliseum spectacles faded due to intervention by the Christian Church, changing societal attitudes, and the empire's declining economy. Beast hunts ended by the 6th century AD because of lack of interest and the difficulty of obtaining exotic animals.
Christian Dirce by Henryk Siemiradzki, 1897, National Museum of Warsaw. The painting depicts a Christian woman killed by a bull as reenactment of the myth of Dirce, with Nero and followers looking on. Dirce was a queen of Thebes who, as punishment for the mistreatment of her niece Antiope, was tied to a wild bull and torn limb from limb. Via Wikimedia Commons and in the public domain.
Environmental Damage
The Roman Empire wiped out entire species from across North Africa and the Middle East, including lions, elephants, and giraffes because of the peoples' addiction to wild animal spectacles. To give one example, during a celebration in the time of Augustus Caesar, 3,500 elephants were killed in a single day.
Most animals transported to Rome from distant provinces didn't survive the journey. Those that did survive died in the arena.
The Roman mania for a plant called silphium that reliably prevented pregnancy, among other uses, caused the plant to go extinct (although botanists still hope to one day stumble upon a few specimens growing in the wilds of North Africa, where it was first discovered).
Roman-era coin with Silphium depicted, via
Wikimedia Commons; in the public domain
The Italian peninsula was essentially deforested early in Rome's development since enormous amounts of wood was needed to fuel the empire's industry, military, and recreation. That deforestation continued as Rome expanded its empire.
Deforestation around the Tiber River and the city of Rome caused the development of malarial marshes, leading to the well-known 'Roman Fever' or malaria that occurred in the region every summer. In the coastal area south of Rome, deforestation created the infamous Pontine Marshes, known for being dangerously malarial through the ages until they were finally drained in the early 20th century.
Massive deforestation by the Romans across the European continent may have triggered the climate change that brought severe drought and famine to Asia in the 4th century. That drought unleashed the invasion of Hunnic warriors into Europe, China, and India, destabilizing the Roman, Han, and Gupta empires respectively, triggering a huge migration of Europeans into areas as far away as North Africa.
Roman-induced climate change may have also unleashed a series of plagues on the ancient world, most notably the Antonine Plague (AD 165-180), the Plague of Cyprian (AD 249-262), and the Plague of Justinian (AD 541-542).
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