Many young adults have not had the MMR as they were too old to have been routinely scheduled for the triple measles, mumps and rubella vaccine when it was introduced in 1988. Other students may only have had one dose, whereas the current recommendation is for two. Plus, it is claimed students are at increased risk of catching mumps particularly because they are in such close contact with so many other young people. 
As with most other infectious disease, measles is most common and most serious in underdeveloped countries. In prosperous first-world countries like the UK and USA, it is rarely a serious disease. In the USA in 2009, there were just two deaths attributed to measles; zero in 2008; zero in 2007 and 2006; and one in 2005. 
Complications from measles are most likely to happen in developing countries where people are vitamin A deficient and serious malnutrition is common. 
Mumps is a viral infection most common in children aged 5-12 and those in crowded settings such as day-cares. Symptoms include headache, muscle aches, tiredness, and loss of appetite. Serious complications from mumps are extremely rare. Very uncommonly, it can cause sterility in post-pubertal males. It is almost never fatal. 
Rubella is a mild viral infection, and between 20 and 50% of cases have no symptoms at all.  It is only mildly contagious, and very uncommon. Between 2009 and 2013, US data shows an average of just six cases annually. 
To sum, these diseases are not only rare in first-world countries, they are also almost never serious. Healthy, well-nourished young people are at very, very low risk of suffering any serious complications in the unlikely event they should contract measles, mumps or rubella. Further, the MMR- as all vaccines - does not confer full or lasting immunity to everyone who receives it. There are many studies and reports documenting outbreaks of disease in highly vaccinated populations.  
Risks of the vaccine:
MMR is one of the most often incriminated vaccine where it comes to reports of serious adverse effects. The manufacturer warns possible adverse reactions to MMR vaccine can include:
As of December 2014, there have been over 6,000 reports of adverse reactions to MMR reported via VAERS, including 329 deaths. 
MMR II contains:
Culture media: Human diploid tissue culture (WI-38), Medium 199
MMRV (ProQuad) contains:
Culture media: Human diploid tissue cultures (MRC-5, WI-38), Medium 199
Bovine calf serum;
Dibasic potassium phosphate;
Dibasic sodium phosphate;
Human serum albumin;
Monobasic potassium phosphate;
MRC-5 cellular protein;
Can I reduce my risk of getting measles, mumps and rubella without getting the vaccine?
Yes. As for all infectious disease, you reduce your chances of acquiring them - or suffering serious complications if you do - by practicing good nutrition and hygiene, and getting plenty of fresh air, sunlight and sleep. If you do acquire measles, an excellent treatment is vitamin A supplements. 
Mumps is mainly transmitted through saliva, so, although unlikely, it can be passed on by sharing utensils with an infected person. Therefore, be careful to use only your own utensils when living in shared accommodation, and thoroughly wash all kitchen utensils before and after use. Mumps can also be transmitted by an infected person sneezing or coughing near you, so ensure everyone you have close contact with knows to cover their mouth when coughing or sneezing. Finally, one of the most common ways mumps is transmitted is through kissing - so be careful who you kiss! 
Rubella is only mildly contagious, but still, as with all disease, to reduce your risk, practice good health habits and avoid things that may suppress your immune system, such as foods high in sugar, highly processed foods, and foods high in hydrogenated oils. 
STRIVE opinion - is it worth getting this vaccine?
do not advise it. The relative risk of a healthy young person acquiring
measles, mumps or rubella is very low, and even if acquired, the
diseases are very rarely serious. Conversely, the vaccine is linked with
a wide and alarming range of negative health outcomes, as well as
raising ethical concerns around its use of animal and human derivatives.
Vegans, vegetarians and people with pro-life beliefs may all wish to
avoid this vaccine, given it contains animal parts, and is grown in a
cell culture derived from aborted foetuses . As with all vaccines, we recommend you do as much research as possible, read the package insert (linked below), and talk to your healthcare professional before reaching a decision.