The contest for technological supremacy, paving the way to future wars of algorithms. Across the skies of Ukraine and Russia, an arms race is unfolding in real time. Earlier this summer it seemed Ukraine had the edge when it delivered drone strikes deep inside Russian territory. But in recent months, Russia has hit back, deploying faster, jet-powered drones, and shifting the balance once again. The Global Jigsaw explores the relentless seesaw battle for technological dominance and ask what it tells us about the future of warfare. Find out more about BBC Monitoring on our website: https://monitoring.bbc.co.uk/ Contributors: Tim Bowler, Margaryta Maliukova, Daria Mosolova Producer: Kriszta Satori Presenter: Krassi Ivanova Twigg Music: Pete Cunningham

To celebrate 10 years of People Fixing the World, Myra Anubi sits down with three regular contributors to talk about the ways the world has changed for the better over the last decade. While rarely making the headlines, huge improvements in healthcare, sanitation and the protection of the environment have taken place since 2016. The group also make their predictions for what positive shifts the world is likely to see in the next 10 years.

How was Jeffrey Epstein able to continue abusing young women for years after he became a convicted sex offender? BBC Eye has been speaking to survivors and analysing emails and documents in the Epstein Files to build up a picture of how a network of contacts in the modelling industry made it possible for the millionaire financier to groom and traffick victims. But will those who helped facilitate Epstein’s crimes ever face justice?

This episode contains references to sexual assault.

Derna, in eastern Libya, was devastated three years ago by the world’s deadliest flood disaster this century. About 4,000 lives were lost. Now, the historic Mediterranean port famous for its fragrant jasmine flowers has been rebuilt, with new roads, bridges, housing blocks, schools and medical facilities. Families that lost their homes have been rehoused, many in spacious apartments. The work’s been planned and paid for by a reconstruction fund with a budget of $2bn, headed by Belgacem Haftar, a son of the military strongman who rules eastern Libya. Reporter Tim Whewell visits Derna to talk to survivors about their memories of the flood, and how they live now. How has the reconstruction been achieved? How does it fit into the political plans of the powerful Haftar family? And what mental scars remain after the flood in a city where almost everyone lost someone they loved, and many bodies have still not been identified? Produced and presented by Tim Whewell

Europe has long relied on America's tech giants to power its digital economy. Now, as geopolitical tensions rise, it is seeking greater control over its digital future through a push for digital sovereignty. We examine what is driving the push and what it could mean for businesses and consumers.

After the Democratic Party swept to victory in the US midterm elections, we bring people together to share their experiences of one of the major issues for voters - the cost of living. When he was elected as president two years ago, Donald Trump pledged to reduce prices. But the opposite has happened. Household costs have gone up, especially for food and fuel. His decision to impose tariffs on some imported goods has made them more expensive. But much of the increase has been driven by a jump in the price of petrol and diesel due to America and Israel’s conflict with Iran. We hear from three young mothers in different parts of the world’s largest economy about their daily struggle to make ends meet. Single mother Lauren from New York tells us she often does not eat properly. "I'm borrowing from Peter to pay Paul because there's nothing left," she says.

In 2005, Michelle Vanek, a 35-year-old mother of four, disappeared on Colorado's Mount of the Holy Cross, a rugged peak so steeped in symbolism that its snow-filled cross has become both landmark and legend. She was hiking with a friend when the pair split up and she vanished. Despite one of the largest search operations in state history, involving helicopters, dogs and hundreds of rescuers, no trace of her was found. But in 2024, an unexpected new chapter unfolded. Pastor Scott Beebe, an experienced search-and-rescue volunteer, dreamt that Michelle came to him, asking to be found "by women." Taking the dream as a kind of calling, he inspired an all-women rescue team to return to the mountain, guided by intuition, faith and a quiet determination to find her and bring her home. Jo Meek speaks to Michelle's husband, Ben Vanek, and to Pastor Scott Beebe, and meets Emily Brown and Erika German from the Vail Mountain Rescue Team who felt a closeness to Michelle on that mountain, decades on.

When cities industrialise the rivers they were built around are first polluted, then turned into sewers and finally built over. It’s a pattern that’s repeated itself for 250 years from 18th century Paris to 21st century Manila. But across the world the fight is on to release and revive our urban rivers. They provide a cooling effect for overheated neighbourhoods, space for wildlife and a welcome place to swim or just dip your toes. Wild swimmer, rower and physicist, Helen Czerski meets the river lovers and the natural life that returns once the light is let back in.

What will the food of the future taste like? In Japan, we discover a fungus turning waste into a high-protein alternative to meat. Then we visit a bedroom where amateur scientists are growing chicken cells. And a restaurant with its very own cultured-meat laboratory upstairs. Could these ideas eventually change the way Japan - and the world - eats food?

Gianni Agnelli is the head of the family that controls Fiat, and the vast business empire beyond the carmaker that helped shape modern Italy. Charming, charismatic and powerful, he's regarded by many as Italy’s unofficial king, and his family de facto royalty. When he dies the country is plunged into mourning and fear for its economic future. But Gianni’s death also triggers an inheritance dispute that will shake his family to its core.

Search for Inheritance wherever you get your BBC podcasts.

Why does artificial intelligence sometimes make things up? From law firms to consultants, businesses are grappling with the risks of AI hallucinations. What's causing them, and can they be prevented?

(Picture: Conceptual image of a large glowing pink holographic cube with "AI" text, positioned as a central hub within a vast, sparse crowd of silhouetted human figures.)