Ole Kirk Christiansen founded the business on 10 August 1932. He was a carpenter in Billund, a village on the flat sandy plain of central Jutland, and the Great Depression had gutted demand for the household goods that had been his living. Toys were what a man with a workshop and no orders could still sell. In 1934 he gave the operation a name built out of the Danish phrase leg godt, play well: Lego. The Latin coincidence was noticed only afterwards, and it is the sort of accident a company is entitled to enjoy, since lego there means roughly I gather together, I assemble, I read. For fifteen years the product was wood.

In 1947 the company bought a plastic injection-moulding machine. Along with it came sample bricks - Self-Locking Building Bricks, invented in 1939 by the English designer Hilary Fisher Page and patented in Britain the following year. Lego's own version, the Automatic Binding Brick, arrived in 1949, and it was not very good. It perched on the studs beneath it and slid off them just as easily. Godtfred Kirk Christiansen, Ole Kirk's son and by 1954 the firm's junior managing director, saw what his father's bricks were failing to deliver: not a toy but a system, in which every piece ever made would fit every piece made since. The tubes were the answer, and the patent went in at the Danish office on 28 January 1958. A fire in the woodworking department ended whatever argument remained. Godtfred stopped wooden production and committed the company entirely to plastic.

What protects the system now is precision. Finished bricks are held to a tolerance of ten micrometres, roughly a tenth the width of a human hair, because a brick slightly too loose falls off and a brick slightly too tight will not come apart. Somewhere near 36 billion elements leave the moulds each year, about 1,140 every second. The last of the original patents expired in 1989 and imitators appeared at once; Lego has since fought them in courtrooms from Beijing to Hamburg with mixed results. Its attempt to trademark the studded look of the brick itself collapsed in Canada, where the Supreme Court held unanimously in October 2005 that trademark law should not be used to perpetuate monopoly rights enjoyed under now-expired patents. Nothing shields the shape any more except the difficulty of making it properly.

By 2003 the company was in genuine danger. It closed the year with a deficit of 1.4 billion kroner, about 220 million US dollars, and Poul Plougmann was replaced as president by Kjeld Kirk Kristiansen, the founder's grandson. Nearly a thousand people lost their jobs in the round of cuts that followed. Then it got worse: the 2004 accounts recorded a net loss of 1,931 million kroner. In October of that year Kristiansen stepped down as president himself, putting 800 million kroner of his own money into the company on his way out, and Jorgen Vig Knudstorp took over as chief executive. The recovery was built almost entirely out of subtraction. Expenditure was cut, a Swiss factory went, and in July 2005 the Legoland parks were sold to Merlin Entertainments for 375 million euros. Revenue rose 12 per cent that year and the company was profitable again.

The village has an international airport because a toy company needed one - the Christiansens began building it in 1961 to move product to the world. By the first half of 2015 Lego had passed Mattel to become the largest toy company on earth by revenue, on sales of 2.1 billion dollars. The line has not run straight since. In September 2017 the group announced 1,400 job cuts, eight per cent of its workforce, after its first revenue decline in thirteen years, competing now not only against Hasbro and Mattel but against every screen in a child's hand. It remains privately held, run by the Kristiansen family through their investment firm Kirkbi since 1995, with Niels B. Christiansen as chief executive and 31,282 employees on the books. The bricks are still made to ten micrometres.