THREE weeks ago, Sarajevo was enshrouded in a cloud of hopelessness. Under daily bombardment by Serb gunners, Sarajevans were convinced they were forgotten by the world. They silently braced themselves for another winter under siege. But two weeks of NATO air raids on Bosnian Serb military positions changed that. Sarajevans now cluster in front of their TVs to see their Army taking Serb-held towns and jubilant government soldiers returning home in flower-covered tanks, instead of the gruesome footage of Bosnian Serb attacks against their capital. Banner headlines in Sarajevo's newspaper Oslobodjenje refer to a Bosnian mistral blowing across the Adriatic Sea - a metaphor for the capture of at least nine Serb-held towns in the past two weeks. ''Bosnian Army takes Kljuc and is advancing toward Sanski Most, Prijedor, and Banja Luka. The Fifth and Seventh Corps are finally going to liberate the country,'' the celebrated paper proclaims. In the past three weeks, the Muslim-led Bosnian Army, aided by its Croat allies, capitalized on the disarray the NATO attacks caused Bosnian Serbs and seized more than 1,500 square miles of land the Serbs had held throughout the 41-month war. The UN estimates the Muslim-Croat federation now holds about 50 percent of Bosnia. Accompanying the military advances on the battlefield and the newly mustered international resolve to end the bloody conflict is the gradual lifting of the siege on Sarajevo. Attacks against the capital have ceased. The Serbs were forced by NATO and the UN to move their heavy guns out of the 12.5-mile exclusion zone. As cafes and restaurants reopen, the streets have come alive. Desolate all summer long, they are now packed with the young, strolling from cafe to cafe. Even a disco has reopened. The first airplanes in five months began landing in Sarajevo amid cheering crowds. The roar of planes hitting the dusty Tarmac brought residents in the surrounding apartment buildings to the streets. ''It was such a welcome sound,'' says Sabina Dzako, sitting on the steps of a nearby apartment in the often-shelled neighborhood of Dobrinja. ''One just feels so differently now. I hope the siege is over, and we will be able to sleep soundly.'' The streets of the Dobrinja neighborhood, which have been empty and silent except for the sound of artillery and machine-gun fire, are now lined with commercial trucks loaded with goods driving over newly opened roads to the capital. Children run among the vehicles, asking for bon-bons. ''It's finished! This is the end of the war,'' says Medzed Muzaferija, who has four shops in Sarajevo where he sells everything from food to sporting equipment. ''Business has improved, and the costs are plummeting. It's not dangerous to walk through Sarajevo anymore.'' City shops, whose barren shelves offered only a few expensive goods three weeks ago, are now brimming with goods, and store owners estimate prices have dropped at least 35 percent. In perhaps the most promising sign of better times ahead, a Benetton shop opened just 50 yards away from a table of flowers commemorating the site of the market massacre on Aug. 28 that killed 39 people. Although few in this war-ravaged city - where the average monthly salary is only about $16 - can afford the pricey Italian clothing, residents welcome the shop. ''It shows this town still lives,'' says Svjetlana Malic, a teenager who works at a city cafe. But despite the fledgling signs that the siege is drawing to an end, not all residents are optimistic. The capital is still without running water, gas, or electricity. To restore electricity and water, both sides must cooperate, because each controls a function necessary for their revival. War-weary residents still cart buckets of water through the city streets and live in darkness as soon as the sun sets. Droves of unemployed people still make meager livings cutting down trees in front-line areas for firewood, and most in the city depend on some form of humanitarian aid.