Particularly in the sprawling capital, Mexico City, known locally as the Federal District, the police have long been considered as much of a threat to the public as the regular professional criminals.
Rape and torture are common, corruption endemic. Eliciting mordidas (bribes) instead of issuing tickets has always been seen as a perk for the poorly paid, badly trained men - and now women - on the beat.
So common was the system that the police rarely had to ask. Mexicans would pre-empt the bribe request with a phrase known as the pa'chesco: "Officer, perhaps we could settle this on the spot. Could I offer you and your partner a refreshment?"
Now, after a record year for crime in the capital - an average 600 serious crimes a day, the worst since the 1910-17 revolution - President Ernesto Zedillo has taken drastic action. He sacked the city's civilian police chief and replaced him with one of the country's toughest army officers, General Enrique Salgado Cordero.
General Salgado is not the first military officer to run the capital's 30,000-member police in times of soaring crime. But his appointment of a dozen other generals and nine colonels to head major precincts, each bringing in their own military staff, was unprecedented. It brought strong criticism, mostly from leftist opposition politicians, that President Zedillo was "militarising" the city of more than 20 million people.
At his first press briefing this week, in a brand new police uniform but with army insignia, General Salgado said he had found the city police "in a state of virtual abandon" and riddled with corruption from the top down.
"I found a force with a wrong and selfish attitude and a tendency towards extortion, often because of exploitation by corrupt senior officers," he said. That appeared to be a reference to the "pyramid" system under which beat and patrol officers - currently earning around 24,000 pesos (about pounds 2,000 a year) - pay what amount to bribes to their superiors, who in turn pay theirs, making the senior officer a very wealthy man.
Beat officers, for example, often have to pay for their own uniforms and hire their own guns, even the bullets, on a daily basis from the sergeant. Those on patrol have to pay "rental" for their cars, the assumption being that their day's bribes will easily cover the cost.
Paralleling Mexico's political system, senior police officers traditionally "bought" the most lucrative posts, an example which filtered down to the lower ranks. A traffic officer, for example, would pay for a particularly rewarding spot at a confusing traffic junction where tourists were liable to miss the red light.
Apart from the almost-daily bank robberies, when police bank guards often happen to be looking the other way, Mexico City residents are concerned by spiralling rates of mugging and highway robbery. Expensive watches are snatched from female drivers in residential areas. The less wealthy are robbed by armed gangs who board the city's "microbuses".
Neither papal nor presidential connections bring immunity. The papal nuncio, Jeronimo Prigione, had his car stolen. President Zedillo's son escaped an attempted kidnap or robbery. Last October, while a committee was debating crime, six armed men strolled into the Mexican parliament building and walked out with the parliamentary staff payroll - around pounds 150,000.
General Salgado pledged to slash the crime figures by next year, saying a "blitz" on police corruption would be the first step. As part of a 22- point anti-crime plan, he said that a data bank, similar to those used to track criminals, would be created to list the record of every police officer. Those with persistent complaints against them would be kicked off the force or prosecuted.
The general also said that he would take hundreds of officers off bank sentry duty to patrol tourist zones or crime-ridden suburbs. It was not uncommon to see a police bank guard prop his shotgun against a tree while parking or guarding customers' cars. For a small refreshment, of course.