Crime works the same way as Disease with regards to decay and diffusion, with the only exception that the population effect is higher (Nightmare/Prince generate 4 crime per pop, Nightmare/Emperor 5 per pop)
As pop effect and sources of crime are usually higher than for the other properties (typically five times), expect your Crime rate to reach much higher (or lower) values.
When Crime reaches a certain level in your city, new "Crime" buildings will appear - Murder, Pickpocket, etc. Most of those buildings are unlocked at specific techs (not just by Crime level) so you may find yourself with a big hit on your economy after researching a tech due to a range of Crime building suddenly appearing.
There are many, many crime building with various prerequisites, maluses and even bonuses; the list below just gives the main effects for all crimes (sorted chronologically by unlock tech) to keep things readable.
Prereq tech|Min Crime|Crime|
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|515|Crime (Kidnapping)||-3|||||||
|525|Crime (Rape)|-2|-3|||||||
|700|Crime (Murder)|-6|-3|||||||
Scavenging|275|Crime (Mugging)|-3|-2|-10||||||
Shelter Building|325|Crime (Burglary)||-2|-5||||||
Cultural Identity|1350|Crime (Sectarian Violence)|-40|-20|||||||
Personal Adornment|35|Crime (Pickpocketing)||-1|-1||||||
Personal Adornment|340|Crime (Identity Theft)||-1|-15||||||
Barter|160|Crime (Shoplifting)||-1|-2||||||
Warfare|300|Crime (Fighting)|-3|-2|||||||
Barter|475|Crime (Bribery)||-2|||||||-20
Warfare|500|Crime (Gangs)|-5|-3|-10||||||
Warfare|920|Crime (Looting)||-3|-50|-20|||||
Warfare|1050|Crime (Rioting)|-20|-8|-50|-30|-30||||
Warfare|1100|Crime (Gang Warfare)|-12|-4|||||||
Fire Making|250|Crime (Arson)|-2|-2|||||||
Chiefdom|1070|Crime (Power Struggles)|-2|-4|||||||
Conduct|725|Crime (Corruption)||-3|-20||||||-30
Trade|1|Crime (Soliciting)||-2|||||||
Fermentation|30|Crime (Public Intoxication)||-1|||||||
Trade|425|Crime (Black Markets)|-4|-2||-10|-20||||
Oratory|25|Crime (False Accusations)||-1|||||||
Writing|75|Crime (Forgery)||-1|-5||||||
Currency|70|Crime (Blackmail)||-1|||||||
Currency|100|Crime (Counterfeiting)||-1|-15||||||
Currency|140|Crime (Extortion)||-2|||||||
Alphabet|175|Crime (Illegal Immigration)||-2||||||-10|
Military Training|460|Crime (Military Desertion)||-2|||||||
Currency|1320|Crime (Rogue Mercenaries)||-4|||||||
Code of Laws|500|Crime (Vigilantism)||1|-5||||||-20
Drug Trade|450|Crime (Drugs)|-5|-2|-30||||||
Drug Trade|1180|Crime (Drug Cartels)|-30|-12|||||||
Sanitation|10|Crime (Littering)|-1|-1|-1||||||
Aesthetics|20|Crime (Graffiti)||-1|||||||
Insurance|120|Crime (Insurance Fraud)||-1|-10||||||
Papacy|42|Crime (Flashing)||-1|||||||
Usury|90|Crime (Embezzlement)||-1|-10||||||
Usury|180|Crime (Tax Evasion)||-1|-20||||||
Surveying|75|Crime (Tresspassing)||-1|||||||
Banking|350|Crime (Bank Robbery)|-2|-1|-25||||||
Matchlock|620|Crime (Armed Communities)|-3|-1|||||||
Corporation|300|Crime (Money Laundering)||-2|-25||||||
Corporation|530|Crime (Corporate Manslaughter)|-6|-2|-15||||||
Stock Brokering|220|Crime (Insider Trading)||-2|||-10||||
Stock Brokering|550|Crime (Stock Market Rigging)||-2|-50||||||
Constitution|900|Crime (Prison Breakouts)||-3|||||||
Emancipation|1000|Crime (Human Trafficking)|-4|-6|||||||
Military Tradition|450|Crime (Insubordination)||-2|||||||
Wildlife Conservation|600|Crime (Poaching)|-2|-2|||||||
Modern Sanitation|10|Crime (Toilet Papering)||-1|||||||
Labor Union|60|Crime (Welfare Fraud)||-1|-5||||||
Labor Union|310|Crime (Racketeering)||-2|||-5||||
Labor Union|520|Crime (Child Labor)|-1|-2|||||-15|-15|
Semi-Automatic Weapons|840|Crime (Gun Rampage)|-3|-6|||||||
Women's Suffrage|50|Crime (Sexual Harassment)||-1|||||||
Women's Suffrage|240|Crime (Domestic Violence)|-1|-2|||||||
Motorized Transportation|50|Crime (Drunk Driving)|-1|-1|-5||||||
Motorized Transportation|130|Crime (Car Chase)|-1|-1|-5||||||
Motorized Transportation|400|Crime (Car Theft)||-2|-15||||||
Motorized Transportation|1125|Crime (Car Bombing)|-3|-4|-5||||||
Compulsory Education|55|Crime (Truancy)||-1|||||-5||
Flight|1300|Crime (Aircraft Hijacking)|-4|-8|-10||||||
Automatic Weapons|650|Crime (Weapons Trafficking)|-6|-1|||||||
Automatic Weapons|950|Crime (Organized Crime)|-8|-6|-10||||||-20
Applied Economics|410|Crime (Ponzi Schemes)||-2|-30||-10||||
Modern Health Care|190|Crime (Elder Abuse)|-3|-1|||||||
Modern Health Care|560|Crime (Medical Fraud)|-6|-2|-10||||||
Modern Health Care|580|Crime (Organ Trafficking)|-6|-1|||||||
Personal Computers|150|Crime (Hacking)||-1||||-5||-5|-5
Minority Rights|250|Crime (Hate Crime)|-2|-2||||||-10|
Ecology|200|Crime (Animal Abuse)||-1|||||||
Ecology|600|Crime (Environmental Dumping)|-8|-3|||||||
Modern Sports|350|Crime (Sports Riots)|-4|-2|-10||||||
Fiber Optics|15|Crime (Online Piracy)||1|-2||||||
Fiber Optics|100|Crime (Internet Fraud)||-1|-10||||||
Genetics|380|Crime (Gene Forgery)||-2|-15||||||
Urban Culture|780|Crime (School Shootings)|-3|-4|||||||
Cloud Computing|15|Crime (Malware)||-1|-10||||-5||-20
Cloning|870|Crime (Unlicensed Cloning)|-2|-4|||||-10||
Online Community|10|Crime (Cyberbullying)||-1|||||||
Cyberwarfare|820|Crime (Cyberwarfare)||-2|-40||||-20||-100
Personal Robots|750|Crime (Asimov Tampering)|-1|-2|||||||
Mind Uploading|1280|Crime (Illegal Brainwashing)|-2|-6|||||||-200
Time Travel|1500|Crime (Timeline Tampering)||-20||||-50|-50|-50|-50