The green zeroes on a roulette wheel are what give the casino a statistical 'house edge', thereby ensuring their continued profitability. Over the years, several strategies have been developed in an attempt to overcome house edge and give players a statistical advantage over the house. It should be noted at this point that none of these strategies have been proven to work reliably, and in many cases are less effective than betting with no strategy at all. The main problem with betting strategies is that past results have no bearing at all on future ones, and to believe otherwise is mere superstition. However, if you are indeed on a lucky streak, and this does happen from time to time, then some of these strategies can multiply your potential winnings.
The most famous roulette strategy is the Martingale system. This involves doubling the bet after every loss, so that the next win will cover all previous losses and make a profit equal to the original bet. The main problem with this strategy is that if the player loses enough times in a row, they will run out of money or reach the table betting limit. If you continue to bet with this system for long periods, then it is a statistical certainty that you will lose a lot of money.
The Fibonacci system is similar in many ways to the Martingale system, except that the bets are calculated using the Fibonacci sequence of numbers, in which the next number is the sum of the previous two. So, the first few numbers of this sequence would be 0,1,1,2,3,5,8 and 13. While this sequence has a certain mathematical beauty to it, and it does have several serious applications in the worlds of science, art, and mathematics, it is largely useless as a betting system, except to say that it is slightly less risky than the Martingale system.
The Labouchere system is another progressive strategy like the Martingale system, but again, it is not quite as risky as it does not require the player to double their bet every time they lose. Instead, it uses a series of numbers in a line to determine the bet amount after a win or a loss. After each bet, the player adds the numbers at either end of the line to determine the size of the next bet. If the player wins, they cross out the numbers at either end, but if they lose, they add the last bet to the end of the line. If the player wins more than a third of their bets, then they should come out ahead.
The D'Alembert System is sometimes referred to as the pyramid system, and is designed to keep both the size of bets and losses to a minimum. After a loss, you add one unit to the next bet, and after a win, you reduce the next bet by one unit. In essence, it is a much more conservative version of the Martingale system, and relies on the same fallacy that a loss is more likely after a win, and vice versa.
While none of these systems are guaranteed to break the bank at your local casino, they certainly add a cerebral element to what is ultimately a game of pure chance, and as long as you don't bet any money that you cannot afford to lose, you will do yourself no harm by employing them.