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Overview: Growth investing is the philosophy of investing in a security that shows signs of above-average earnings growth as compared to its industry or the overall market, even if the security appears expensive from a price-to-earnings or price-to-book perspective.
Theory: In addition to above average earnings growth, the theory behind growth stock investing, as opposed to value investing, is that stocks breaking into new price highs have no overhead supply. Because there is no overhead supply with stocks breaking into new price highs, the stock runs into less resistance. 
People: William O'Neil, who is recognized as the father of growth stock investing dubbed this phenomenon the "Great Market Paradox". O'Neil in his book "How To Make Money In Stocks" claims to have researched the greatest winning stocks, and developed the "CAN SLIM" system that is largely the basis of growth stock investing.
This style of investing is also called capital growth investing since growth investors seek to maximize capital gains, not income from dividends. Companies that generally fall under this category tend to be driven by new technologies and/or domination of a niche market.
Notable proponents of this strategy include Philip Arthur Fisher, Jim Slater, Peter Lynch and Warren Buffett, although the latter has often maintained that there is no theoretical difference between value investing and growth investing.