As well as the sporting heroes who are recognised for their endeavours, honours also go to Star Wars and Trainspotting star Ewan McGregor, actress Siobhan Redmond and violinist Nicola Benedetti.
Professor Peter Higgs, the retired Edinburgh University professor who gave his name to the "God particle", is awarded the Companion of Honour for his services to physics.
Grainger, who won her gold medal after being runner-up at the previous three Games, is made a CBE, while Murray, who took Olympic gold and won his first grand slam when he beat Novak Djokovic in the US Open, is awarded an OBE.
In full: the New Year Honours list
Edinburgh-born rower Sophie Hosking, who took gold in lightweight women's double sculls at this summer's Olympics, becomes an MBE, with the same honour going to horse rider Scott Brash from Peebles, who at 27 was the youngest member of the gold medal-winning British showjumping team at the London 2012.
Another Olympics star to be made an MBE is Aberdeen- born Tim Baillie, who won gold in the canoe slalom C2 event.
Scotland's Paralympic stars are also honoured, with David Smith being made an MBE after he was part of the team that won the mixed coxed four event at the Games.
Cyclist Neil Fachie, who triumphed in the 1km time trial for the blind and visually impaired, is also made an MBE, with the same honour going to cycling pilot Craig MacLean, who was part of the duo that won the tandem sprint at the Paralympics.
Sportscotland chairwoman Louise Martin CBE, said: "These honours are a fitting testament to the incredible achievements of the Scottish Olympians and Paralympians. Their success has helped inspire a nation, and the momentum is continuing as Scotland prepares to host the Commonwealth Games in 2014."
Actor McGregor receives an OBE for his services to drama and charity. The Crieff-born star, an ambassador for Unicef, said he was "delighted and touched" with the accolade.
Redmond is made an MBE for services to drama. The actress, who was born in Tollcross, Glasgow, made her television debut in the early 1980s before going on to star in shows including Taggart and Holby City.
The same honour goes to Benedetti, for her services to music and charity. The 25-year-old, who was born in West Kilbride, began violin lessons at the age of five and was named BBC's Young Musician of the Year in 2004 at 16.
She has worked with organisations such as Sistema Scotland to help demonstrate the power of music to transform young people's lives.
Others who are honoured include John Leighton, the director general of the National Galleries of Scotland, who is knighted for his services to art.
Also receiving knighthoods are Professor Ian Diamond, the principal and vice-chancellor of Aberdeen University, for his services to social science and higher education; and Edinburgh-born Professor Hew Strachan, the Chichele Professor of the History of War at All Souls College, Oxford.
He was previously professor of modern history at Glasgow University and was also director of the Scottish Centre for War Studies.
Orkney jewellery designer Sheila Fleet, who set up her business almost two decades ago, is honoured with an OBE for services to the industry.
Her family-run business, Sheila Fleet Jewellery, has grown to become one of the biggest employers in the rural countryside of Orkney.
Designer Joyce Young, who creates wedding dresses and mother-of-the-bride outfits, is another to receive an OBE.
Ms Young, who launched her company By Storm in 1993 having been a designer in the fashion industry since graduating from Glasgow School of Art in 1975, is being honoured for her services to the textile industry and the community in Glasgow.
The same honour also goes to Dr Bruce Ritson, the chairman of Scottish Health Action on Alcohol Problems, an independent medical organisation set up to raise awareness of the nature and extent of harm linked to alcohol use.
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