The controversy so often associated with the Olympic Games began to reach Winter Olympia in 1968 at Grenoble. Though the Games went fairly well, there were many problems. It began before the Olympics when the IOC, led by President Avery Brundage, decided it wished to curb advertising on skis and clothing by the alpine skiiers. They threatened to expel certain skiiers, while the skiiers threatened to withdraw en masse in revolt. A compromise was reached eventually in which the skiiers agreed to remove all equipment with advertising prior to being photographed or interviewed.
Grenoble was in the Dauphine region of the French Alps, but the 1968 Winter Olympics were spread all through the mountains. There were separate Olympic Villages because the distance to the events – a main one in Grenoble, and the others in Chamrousse and Autran. Only the skating and ice hockey events were actually held in Grenoble.

The Games' hero of heroes was Jean-Claude Killy, who had grown up and learned to ski in the neighboring mountains. He was favored, and expected by France to duplicate Toni Sailer's 1956 feat and win all three alpine skiing gold medals. He succeeded but not without a major controversy in the slalom. The race was held in fog and both Karl Schranz (AUT) and Håkon Mjøen (NOR) posted faster times. But both were disqualified for missing gates. Schranz appealed, stating that he had been interfered with and replays showed he had. He was allowed a re-start and again posted a winning time. But he was then disqualified when further investigation revealed that the interference on the first run occurred after Schranz had missed his gates. Killy had his third gold medal and France had its hero.
Another star of the Games was America's Peggy Fleming. She was the American figure skating hope to resurrect the American program after the tragic events of February 1961, when the entire American figure skating was killed in a plane crash en route to the world championships. Fleming was the heavy favorite to win the women's figure skating title at Grenoble, having won the world championship in 1966-1967, and she did not disappoint, winning the women's title easily, placing first both in school figures and free skating. In pairs figure skating, the lyrical, almost poetic team of Lyudmila Belousova and Oleg Protopopov won their second consecutive championship.
In the bobsled events, held on l'Alpe d'Huez, site of so much heroism and suffering during the Tour de France, Eugenio Monti of Italy finally succeeded in winning an Olympic gold medal. In fact he won both bobsled events. Monti was a multiple world champion, and had been considered the best bobsledder in the world, but he had not previously been able to strike Olympic gold.
Grenoble was really the first time that controversy marred the Olympic Winter Games, which had been relatively immune to the political problems so often seen at their summer counterparts. It would not get any better at the next few Winter Olympics. Taken from http://www.sports-reference.com/olympics/winter/1968/

In 1963, the IOC awarded the Olympics to Mexico despite some warnings about the effects of competing at the altitude (2,134 metres) of Mexico City. The warnings would prove prophetic, both for good and bad, but also prominent at Mexico City was the first large-scale incursion of politics directly into the Olympic venues.
Political problems manifested themselves as protests by Mexican students before the Games. The students were upset that so much money was spent on the Olympics in the face of such widespread poverty in their country. As the protest movement gathered momentum leading up to the Games, the Mexican army took charge on the night of 2 October. As 10,000 people demonstrated in the Square of the Three Cultures in Mexico City, the army surrounded the crowd and opened fire. More than 250 people were killed and thousands were injured. In the United States, Harry Edwards, a professor at San Jose State University, urged blacks to boycott the Olympics to protest the rampant racism of American society. The boycott never materialized. However, his efforts came to fruition in the victory ceremony of the 200 metres. The race was won by Tommie Smith (USA) with the bronze medal going to John Carlos (USA). On the victory platform, as the Star Spangled Banner played in the background, almost unheard, the two black Americans stood barefooted, heads bowed, and raised a singled black-gloved fist in their own form of protest. The IOC banned the two from future Olympic participation and ordered them to leave the Olympic village immediately.

On a positive political note, the Federal Republic of Germany (West Germany) and the German Democratic Republic (East Germany) entered separate national teams for the first time.
The altitude severely affected many track & field events. Bob Beamon used the lesser gravity to set a stunning world record in the long jump of 8.90 (29’2½”). In the 100, 200, 400, 400 metre hurdles, 4×100 relay, 4×400 relay, and the triple jump, all sprint events not requiring much oxygen, and aided by the lessened pull of gravity, new world records were set by the men. Many of these records would not be broken for years.
But the distance running events saw very slow times, as the runners gasped for the oxygen that was not there. In the 5,000 metres, the winning time was the slowest in 16 years. In the marathon, Mamo Wolde (ETH) won in a time eight minutes slower than his countryman, Abebe Bikila, had posted in Tokyo. Bikila ran in Mexico but was hampered by a stress fracture and withdrew at 17 miles.
The two saddest sights attributed to altitude were the men's 1,500 metres and 10,000 metres. In the 1,500 the race was seen as a dual between Kenya's Kip Keino, who had grown up at altitude and trained there regularly, and America's Jim Ryun. Ryun had become the dominant miler in the world in the past few years, though he was slightly less so in 1968. Still, he was no match for Keino in the thin air of Mexico and had to settle for the silver medal. It is possible that Keino would have won at any altitude but there is no way to know.
In the 10,000 metres, Australia's Ron Clarke, one of the finest distance runners ever against the clock, was still searching for his first gold medal. He ran himself into exhaustion, but could finish only sixth. Through 8,000 metres, the lead pack was Clarke and Naftali Temu (KEN), Mamo Wolde (ETH), and Mohamad Gammoudi (TUN) – all altitude-trained Africans. Clarke was dropped at that point and finished 6th. At the finish he collapsed, ashen-faced, and lay unconscious, near death, for ten minutes as the Australian team doctor wept openly while administering to him. Clarke recovered fully, but he would never win a gold medal.
Taken from  http://www.sports-reference.com/olympics/summer/1968/