SAN LUIS, Cuba — In Cuba, there’s only one tobacco grower whose face has been deemed worthy to grace a box of Cuban cigars. It’s the late Alejandro Robaina, and premium Vegas Robaina cigars have spread his fame around the globe.

Now his descendants work the same tobacco lands in Cuba’s famed Vuelta Abajo, where the creme de la creme of Cuban tobacco and the finest capas — wrappers — are produced using the same methods of Don Alejandro. Before he died in 2010, the man who rose from humble roots on the leaves of tobacco had traveled the world as an ambassador for Cuban cigars.

So popular were Vegas Robaina cigars during the XVIII Festival Habano, which celebrated the Cuban cigar from Feb. 29 to March 4 this year, that there was scarcely a box to be found in Havana weeks after the event ended.

The rapprochement between the U.S. and Cuba and the influx of American visitors eager to sample one of Cuba’s most famous products also has spiked demand for Habanos.

“The demand for cigars by tourists is probably higher than it’s ever been. Many people think it’s not a complete trip unless they buy and smoke a Cuban cigar,” said David Savona, executive editor of Cigar Aficionado magazine.

Cuban cigars also have growing cachet in emerging markets like China and Russia, Savona said, but other than some super premium brands, Cuban cigars aren’t currently in short supply because the weak global economy is cutting into sales in some of Cuba’s traditional European markets. Prices have edged up slightly, he said, and Cuba is adding more premium, high-priced cigars to its offerings.

Despite Cuban tobacco’s fame, it isn’t a big contributor to Cuba’s bottom line. Production is limited. It did pick up in 2015, however, with 24,500 metric tons produced, compared to 19,800 metric tons the previous year, according to Cuba’s National Office of Statistics and Information. Land planted in tobacco also was up significantly in 2015 compared to the previous four years.

“When the embargo is lifted, the Cubans will need to up production even more,” Savona said. “But production and sales aren’t always at an equal pace. Cigars do get better with age.”

For the Robainas, tobacco is a way of life that began when their maternal ancestors, the Peredas, came from the Canary Islands in 1845 and settled in the area of western Cuba known as Cuchillas de Barbacoa. It turned out that Cuchillas de Barbacoa with its cool nights, fertile soil and abundant water is the best place in all of Cuba — and perhaps the world — to grow quality tobacco.

Alejandro Robaina, who was born in 1919, began working the family tobacco lands when he was 10 — also the age when he sampled his first cigar — and later took over from his father Maruto. Along the way, he became a master tobacco grower and his wrapper tobacco became legendary.