The Non-Aligned Movement (NAM) is a group of states which are not aligned formally with or against any major power bloc. As of 2012, the movement has 120 members and 17 observer countries.
In a speech given during the Havana Declaration of 1979, Fidel Castro said the purpose of the organization is to ensure "the national independence, sovereignty, territorial integrity and security of non-aligned countries" in their "struggle against imperialism, colonialism, neo-colonialism, racism, and all forms of foreign aggression, occupation, domination, interference or hegemony as well as against great power and bloc politics.
The 16th NAM summit took place in Tehran, Iran from 26 to 31 August 2012. According to MehrNews agency, representatives from over 150 countries are scheduled to attend... At the summit, Iran is taking over from Egypt as Chair of the Non-Aligned Movement for the period 2012 to 2015
Fidel Castro Cuba Communist Party of Cuba 1979-1983
Nelson Mandela South Africa African National Congress 1998-1999
Fidel Castro Cuba Communist Party of Cuba 2006-2008
Raśl Castro Cuba Communist Party of Cuba 2008-2009
Hosni Mubarak Egypt National Democratic Party 2009-2011
Mahmoud Ahmadinejad Iran Alliance of Builders of Islamic Iran 2012 Present
Now that the Mahmoud Ahmadinejad, has become the chairman of the Non-Aligned Movement (NAM), the president of Iran chose to make NAM the center-piece of his speech to the General Assembly of the UN.
Ahmadinejad addresses UN General Assembly (Live cut version)
I chuckle every time this movement makes the news. What started out as a misfit band of Che-inspired, violent dictators in populist clothing...has remained exactly that after 50+ years!
NAM provides no valuable benefit to the world. In fact, it harms the planet by lending legitimacy to the useless regimes that often end up setting its agenda. What's worse than the UN? The UN with only the lamest countries involved.
All of this said, the unfortunate answer to the question is yes - they do have an influence on world events. It's simply by virtue of the time and money spent by these countries on their futile initiatives.