Very few drugs work perfectly, and Viagra is no exception.
The first problem comes because Viagra happens to have a spillover effect. It blocks PDE5, but it also has an effect on PDE6. It turns out that PDE6 is used in the cone cells in the retina, so Viagra can have an effect on color vision. Many people who take Viagra notice a change in the way they perceive green and blue colors, or they see the world with a bluish tinge for several hours. For this reason, pilots can't take Viagra within six hours of a flight.
The second concern comes for people who are taking drugs like nitroglycerin for angina. Nitroglycerin works by increasing nitric oxide, and it helps with angina by opening up the arteries that supply the heart with oxygen. If you take nitroglycerin and Viagra together, the increased nitric oxide plus the blocking of PDE5 can lead to problems.
Other issues with Viagra can include little things like headaches (the drug, as a side effect in some men, opens up arteries in the brain's lining and causes excess pressure) and big things like heart attacks. The possibility of heart attacks is one reason why Viagra is a prescription drug rather than an over-the-counter drug like aspirin. A doctor needs to understand your medical history and make sure that Viagra won't cause a heart attack.
Viagra's most famous possible side effect? A long-lasting (as in, several hours long) and painful erection.
Finally, there's concern that some men -- especially younger men who take Viagra recreationally and who don't really need it for physical reasons -- may end up with a dependency on the drug and become unable to maintain an erection without taking Viagra.