Paul Erdős was born in Budapest, Hungary on March 26, 1913. The only surviving child of Anna and Lajos Erdős, his siblings died before he was born, aged 3 and 5. His parents were both Jewish mathematicians, from a vibrant intellectual community. His fascination with mathematics developed early – at the age of three, he could calculate how many seconds a person had lived. Both of Erdős's parents were high school mathematics teachers, and Erdős received much of his early education from them. Erdős always remembered his parents with great affection. At 16, his father introduced him to two of his lifetime favorite subjects – infinite series and set theory. During high school, Erdős became an ardent solver of the problems proposed each month in KöMaL, the Mathematical and Physical Monthly for Secondary Schools. Erdős later published several articles in it about problems in elementary plane geometry.

In 1934, at the age of 21, he was awarded a doctorate in mathematics. Because anti-Semitism was increasing, he moved that same year to Manchester, England, to be a guest lecturer. In 1938, he accepted his first American position as a scholarship holder at Princeton University. At this time, he began to develop the habit of traveling from campus to campus. He would not stay long in one place and traveled back and forth among mathematical institutions until his death.

Possessions meant little to Erdős; most of his belongings would fit in a suitcase, as dictated by his itinerant lifestyle. Awards and other earnings were generally donated to people in need and various worthy causes. He spent most of his life as a vagabond, traveling between scientific conferences and the homes of colleagues all over the world. He would typically show up at a colleague's doorstep and announce "my brain is open," staying long enough to collaborate on a few papers before moving on a few days later. In many cases, he would ask the current collaborator about whom he (Erdős) should visit next. His working style has been humorously compared to traversing a linked list.

His colleague Alfréd Rényi said, "a mathematician is a machine for turning coffee into theorems", and Erdős drank copious quantities. After 1971 he also took amphetamines, despite the concern of his friends, one of whom (Ron Graham) bet him $500 that he could not stop taking the drug for a month. Erdős won the bet, but complained during his abstinence that mathematics had been set back by a month: "Before, when I looked at a piece of blank paper my mind was filled with ideas. Now all I see is a blank piece of paper." After he won the bet, he promptly resumed his amphetamine habit.

He had his own idiosyncratic vocabulary: he spoke of "The Book", an imaginary book in which God had written down the best and most elegant proofs for mathematical theorems. Lecturing in 1985 he said, "You don't have to believe in God, but you should believe in The Book." He himself doubted the existence of God, whom he called the "Supreme Fascist" (SF). He accused the SF of hiding his socks and Hungarian passports, and of keeping the most elegant mathematical proofs to himself. When he saw a particularly beautiful mathematical proof he would exclaim, "This one's from The Book!". This later inspired a book entitled Proofs from THE BOOK.

Other idiosyncratic elements of Erdős' vocabulary include:

• children were referred to as "epsilons" (because in mathematics, particularly calculus, an arbitrarily small positive quantity is commonly denoted by that Greek letter (ε));
• women were "bosses";
• men were "slaves";
• people who stopped doing math had "died";
• people who physically died had "left";
• alcoholic drinks were "poison";
• music was "noise";
• people who had married were "captured";
• people who had divorced were "liberated";
• to give a mathematical lecture was "to preach" and
• to give an oral exam to a student was "to torture" him/her.

Also, all countries which he thought failed to provide freedom to individuals as long as they did no harm to anyone else were classified as imperialist and given a name that began with a lowercase letter. For example, the U.S. was "samland" (after Uncle Sam), the Soviet Union was "joedom" (after Joseph Stalin), and Israel was "israel". For his epitaph he suggested, "I've finally stopped getting dumber." (Hungarian: "Végre nem butulok tovább").

In 1952, during the McCarthy anti-communist investigations, the U.S. government denied Erdős, a Hungarian citizen, a re-entry visa into the United States, for reasons that have never been fully explained. Teaching at Notre Dame at the time, Erdős could have chosen to remain in the country. Instead, he packed up and left, albeit requesting reconsideration from the Immigration Service at periodic intervals. The government changed its mind in 1963 and Erdős resumed including American universities in his teaching and travels. Hungary, then a Communist nation, was under the hegemony of the Soviet Union. Although it curtailed the freedoms of its citizens, in 1956 it gave Erdős the singular privilege of being allowed to enter and exit Hungary as he pleased. Erdős exiled himself voluntarily from Hungary in 1973 as a principled protest against his country's policy of denying entry to Israelis.

During the last decades of his life, Erdős received at least fifteen honorary doctorates. He became a member of the scientific academies of eight countries, including the U.S. National Academy of Sciences and the UK Royal Society. Shortly before his death, he renounced his honorary degree from the University of Waterloo over what he considered to be unfair treatment of a colleague. He died "in action" of a heart attack on September 20, 1996, at the age of 83, while attending a conference in Warsaw, Poland.

Erdős was one of the most prolific publishers of papers in mathematical history, second only to Leonhard Euler; Erdős published more papers, while Euler published more pages. He wrote around 1,475 mathematical articles in his lifetime, mostly with co-authors. He strongly believed in and practiced mathematics as a social activity, having 511 different collaborators in his lifetime.

In terms of mathematical style, Erdős was much more of a "problem solver" than a "theory developer". Joel Spencer states that "his place in the 20th-century mathematical pantheon is a matter of some controversy because he resolutely concentrated on particular theorems and conjectures throughout his illustrious career." Erdős never won the highest mathematical prize, the Fields Medal, nor did he coauthor a paper with anyone who did, a pattern that extends to other prizes. He did win the Wolf Prize, where his contribution is described as "for his numerous contributions to number theory, combinatorics, probability, set theory and mathematical analysis, and for personally stimulating mathematicians the world over". In contrast, the works of the three winners after were recognized as "outstanding", "classic", and "profound", and the three before as "fundamental" or "seminal".

Of his contributions, the development of Ramsey theory and the application of the probabilistic method especially stand out. Extremal combinatorics owes to him a whole approach, derived in part from the tradition of analytic number theory. Erdős found a proof for Bertrand's postulate which proved to be far neater than Chebyshev's original one. He also discovered an elementary proof for the prime number theorem, along with Atle Selberg, which showed how combinatorics was an efficient method of counting collections. Erdős also contributed to fields in which he had little real interest, such as topology, where he is credited as the first person to give an example of a totally disconnected topological space that is not zero-dimensional.

Throughout his career, Erdős would offer small prizes for solutions to unresolved problems. These ranged from $25 for problems that he felt were just out of the reach of current mathematical thinking, to several thousand dollars for problems that were both difficult to attack and mathematically significant. There are thought to be at least a thousand such outstanding prizes, though there is no official or comprehensive list. These prizes are still active despite Erdős' death – Ron Graham is the (informal) administrator of solutions. Winners can either get a check signed by Erdős (for framing only) or a cashable check from Graham.

Perhaps the most famous of these problems is the Erdős conjecture on arithmetic progressions:

If the sum of the reciprocals of a sequence of integers diverges, then the sequence contains arithmetic progressions of arbitrary length.

If true, it would solve several other open problems in number theory. (Although one main implication of the conjecture, that the primes contain arbitrarily long arithmetic progressions, has since been proved independently as the Green-Tao theorem.) The problem is currently worth US$5000.

Because of his prolific output, friends created the Erdős number as a humorous tribute; Erdős alone was assigned the Erdős number of 0 (for being himself), while his immediate collaborators could claim an Erdős number of 1, their collaborators have Erdős number at most 2, and so on. Approximately 200,000 mathematicians have an assigned Erdős number, and some have estimated that 90 percent of the world's active mathematicians have an Erdős number smaller than 8 (not surprising in light of the small world phenomenon). It is said that Baseball Hall of Famer Hank Aaron has an Erdős number of 1 because they both autographed the same baseball when Emory University awarded them honorary degrees on the same day. Erdős numbers have also been assigned to an infant, a horse, and several actors. The Erdős number was most likely first defined by Casper Goffman, an analyst whose own Erdős number is 1. Goffman published his observations about Erdős' prolific collaboration in a 1969 article entitled "And what is your Erdős number?"