The largest mosaic made from scrambled Rubik's Cubes measured 68 m [220 ft] x 4 m [13 ft]. The mosaic, showing the Macau skyline, was created in December 2012 by  Cube Works Studio from 85,794 cubes. It took 90 days from design to completion. VIDEO

previous records:
10.28 m [33 ft 8 in] x 4.40 m [14 ft 5 in], created at Hotel Loi Suites Iguazú, Puerto Iguazú, Argentina on 8 March 2012.
a replica of the hand of god in Michelangelo's "The Creation of Adam" from the Sistine Chapel in the Vatican made from 12,090 cubes by Cube Works Studio in Toronto (Canada), measuring 8.8 x 4.4 m [29 ft x 14.5 ft].
9,071 cubes,  students of the Machidashiritsu Yamasaki School in Tokyo at the Tokyo Toy Show on 15 July 2010
a replica of "The Last Supper", made of 4,050 cubes by Cube Works Studio in Toronto (Canada) in 2009.
The photo shows the first record in this category, set in Galicia, Spain. Five people needed 5 hours to solve and assemble 1848 cubes.
(see photo, with permission from David Calvo)
MORE DETAILS AND PHOTOS

The individual record was a Christmas tree composed by Bernett Orlando (India) from 2025 cubes in Cologne (Germany) in December 2009 (see photo, more photos can be found HERE.)

The fastest robot to solve a Rubik's Cube is Sub1 Reloaded, constructed by Albert Beer (Germany). It solved the cube in 0.637 seconds on 9 November 2016 at the electronica trade fair in Munich, Germany VIDEO
MultiCuber 3, constructed by David Gilday set a record for completing a 4x4x4 cube by a robot in 1:16.68 minutes on 15 March 2014 at Big Bang Fair in Birmingham.
previous record holders:

0.887 seconds, Sub1, constructed by Albert Beer (Germany), 23 January 2016 at the Cubikon Store in Munich. VIDEO
0.9 seconds by Paul Rose and Jay Flatland (USA) VIDEO
2.4 seconds by Zackary Gromko (USA) at Saint Stephens, Bradenton, Florida, USA, on 15 October 2015 VIDEO
CubeStormer III, developed by Mike Dobson and David Gilday. It solved a Rubik's Cube in 3.253 seconds on 15 March 2014 at Big Bang Fair in Birmingham. VIDEO
CubeStormer II, developed by Mike Dobson and David Gilday solved a Rubik's Cube in 5.27 seconds. DETAILS AND VIDEO
Ruby, developed by Swinburne University students (Australia), 10.18 seconds
Rubot II, developed by Peter Redmond (Ireland), solved a scrambled Rubik's Cube within 64 seconds (including the time to scan the initial position) on 8 January 2009 at the Young Scientist show in the Royal Dublin Society . DETAILS VIDEO

The largest Rubik’s Cube solved by a robot is a 9x9x9 cube successfully solved in 34:25.89 minutes by MultiCuber 999, built by David Gilday (UK), on 15 March 2014 at Big Bang Fair in Birmingham.

The largest Rubik's Cube was built by Tony Fischer (UK). It has an edge length of 1.57m. VIDEO

The most working layers has a 22x22x22 cube.  It has an edge length of 200mm and contains 2691 working parts. The cube can be ordered by Thingverse. VIDEO
The largest mass-produced and commercially available cube is the YuXin 17x17x17 cube.

The smallest working Rubik's Cube is 5.7 mm wide. It was created using a 3D-printer by Evgeniy Grigeriev (Russia). The miniature cube can be obtained here.
VIDEO

The best known algorithm for solving the cube needs only 20 face turns, details can be found at cube20.org. MORE DETAILS

The most expensive Rubik's Cube was the Masterpiece Cube, produced by Diamond Cutters International in 1995. The actual-size, fully functional cube features 22.5 karats of amethyst, 34 karats of rubies, and 34 karats of emeralds, all set in 18-karat gold. It has been valued at about US-$ 1.5 mio.

4x4x4 CUBE

fastest time: 17.42 seconds, Sebastian Weyer (Germany) at the Danish Open 2019 VIDEO
blindfold, fastest time (including memorising): 1:02.51 min, Stanley Chapel (USA) at the Michigan Cubing Club Epsilon 2019

5x5x5 CUBE

fastest time: 34.92  seconds, Max Park (USA) at the Houston Winter tournament 2020
blindfold, fastest time (including memorising): 2:21.62 min, Stanley Chapel (USA), Michigan Cubing Club Epsilon 2019

6x6x6 CUBE

fastest time: 1:09.51 min, Max Park (USA) at the Houston Winter tournament 2020

7x7x7 CUBE

fastest time: 1:40,89 Min, Max Park (USA) at the CubingUSA Nationals 2019