But far removed from TV screens in the UK, another very demanding audience will be closely watching too: in Russia, and in their thousands.
The film-makers may not have them in mind.
But Russian TV has shown strong interest in buying the rights, although nothing is yet confirmed.
Eager Tolstoy aficionados in Russia have found their own ways, through pirated live streams and illegal uploads on social networks.
Unofficial "studios" have even provided single-voice translations with minimal delay.
Quite apart from the legal issues of pirating the show, Russians have been logging on to social media sites and giving their views.
One special group has been set up on Russia's big social media site, VKontakte. The reviews are mainly positive and many are clearly in awe of the British take on one of the most famous works of Russian literature.
But there is withering criticism too.
"Just looking at the actors! I feel their choice is wrong. In the book Natasha is much younger and more playful, like a kid, and here she's a fully formed woman," complains Yulia Drutsko "It's a Hollywood film that's incapable of carrying across the message of the book."
And it's not just her age. Natasha Rostova (played by Lily James) also takes a lot of Russian flak for being blonde.
Then there is the fashion. "You need to hack the hands off those who thought of those miserable costumes," says Alyona Reshetnikova. "You can see instantly those are not Russian people. Only Pierre (Paul Dano) saves the situation."
"Adding debauchery is wrong. If Europeans can't take this novel without erotic scenes, they should have left the book alone. I think Tolstoy's book is wonderful without all that," adds Polina Krasnova, echoing the popular view that the creators injected way too much carnality, for the sake of driving up viewing numbers.
Tom Harper, the director of the series, told the BBC he was "delighted that people in Russia are interested in watching it" and said that the racier scenes were a part of the "essence" of the original text.
"Plenty of people have said that Tolstoy wasn't able to write these things and that this series works with the essence of what's in the book, but it's such a small part of the adaptation overall," he said.
At least one Muscovite, Kristina Kim, was not won over by the adaptation and gave up halfway through. "Pigs running in the middle of Moscow, Countess Mary's clothes and hair done like a maid, and then the Husssar's pelisse slung over the right shoulder? Don't even get me started on the battle scenes," she complained.
Ask any Russian, and they will of course tell you your pelisse goes over the left shoulder. Sling it over the right and you will impede sabre-rattling. Obviously.
In her disappointment, Kristina was driven to watch the 1967 Soviet epic that won director Sergei Bondarchuk an Oscar.
"The interiors, the decor, the Moscow winter and the war scenes especially are ideal there," she told the BBC. "Of course Field Marshal Kutuzov and all the fighting were shot with such expense you wouldn't expect any flaws."
No effort and no expense were spared in providing the Soviet film-makers with what they needed.
- 3,000 Soviet soldiers were drafted for one battle scene
- 57 museums donated exhibits for the shoots
- More than 40 state firms were enlisted to produce replica weapons and costumes
- It took seven years to make and cost millions of roubles, at the time an astonishing sum
This was because the Soviet film had not only cinematic, but political importance.
The authorities in Moscow were anxious that Hollywood had got there first, with the 1956 film featuring Audrey Hepburn and Henry Fonda.
Such an important task could not be entrusted to young actors, so Soviet film-goers saw 20-year-old Pierre Bezukhov played by Sergei Bondarchuk himself, then 47. And Vyacheslav Tikhonov, who played Prince Andrei Bolkonsky, was nearly 40.
At 25, the Soviet Natasha, actress Ludmila Savelyeva, came close to the age Leo Tolstoy imagined her to be.
Now 74, she believes that it is extremely hard to get the cast 100% right in a film with so many actors.
"I saw the 2007 [Franco-Italian-German-Russian] TV production and the cast was mainly a mish-mash, although one male actor was very nice," she told BBC. "Their Natasha was acting really strange, running barefoot and hanging on Pierre's neck."
She is waiting eagerly for the BBC series to be shown on Russian TV.
Whatever Russians feel about the latest adaptation, it has revived their interest in the story and sent many back to Tolstoy's 1,300-page original.
Every 16-year-old in Russia is required to read the epic work under the school curriculum, and many Russians still remember having to recite Prince Andrei's three pages of thoughts under an oak tree.
"I've read up to where Andrei was wounded: should I continue watching?" asks an engrossed Dmitry Nuritov, 18, from Chelyabinsk.
"I think I'll read the book first," he decides.
