Is sport betting legal in Europe or America?

Sport betting in most of the world usually prohibited by law but each country or state has it own rules and regulations regarding betting and can choose to make it legal.

But this vast industry of sports betting has found ways to survive in an orderly way by some authorities.
The most familiar way is a government gambling authority. The main purpose of these government agencies is to allow citizens to gamble legally, this because the profits are directed mostly for public service or state. For example, support the establishment of Sports Toto and Lotto (National Lottery) construction of public buildings (schools, libraries, etc.).

These legal sports betting usually has small chance of winning and this is the reason that many people prefer to bet with illicit bookmakers (known colloquially as "bookies") or sports online betting.

In the United States of America, it is illegal to operate a betting scheme, except in a few states. In many European nations, bookmaking (the profession of accepting sports wagers) is regulated but not criminalized.

Among the areas which gambling is legal are: Nevada(Las Vegas), Atlantic City, New Jersey or Tunica, Mississippi.

I agree with you that "betting may give rise to match-fixing" but I see this as a reason to make gambling legal and supervised because let's face it gambling will occur anyway.
It better to address the money to the country and community needs rather than today when in the money goes to private people or worse to fund terror, drugs and other illegal activities.

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In the UK we have various ways to gamble on sports - many bookmakers allow bets on horse racing, football, rugby, whether Christmas will have snow, whether a meteorite will fall out of the sky - in short, pretty much anything.

This is a very strongly regulated industry - with a range of checks to ensure gambling rings don't influence individual players or teams.