Do astronauts age while traveling in space?
1 Answer
Yes  they will experience aging in what they perceive as a normal way. An Earthbased observer will observe an astronaut aging slower or faster depending on various factors discussed below.
Explanation:
Do astronauts age while traveling in space? Let's explore that question.
The first thing to keep in mind is that an astronaut will experience time normally  24 hours will be experienced as 24 hours (and not something shorter or longer). And so from the astronaut's point of view, s/he will age and age normally.
It's when we start comparing the aging rate to someone in a different location, like an observer on Earth, that things start to get complicated.
There are two ways that time can dilate  that is, that time can be seen to be passing at a slower rate for different observers:

High velocity. Einstein's Theory of Special Relativity says that time will dilate (slow down) for someone experiencing high velocity. Astronauts experience a very high velocity compared to an observer on Earth and so the Earth observer will observe that the astronaut ages more slowly.

High gravity. Einstein's Theory of General Relativity says that time will dilate for someone in a high gravitational field. Astronauts experience lower gravity than on Earth and so the Earth observer will observe that the astronaut ages more quickly.
I should note that for both of these effects, because we're not working with extreme velocity (a significant fraction of the speed of light) or extreme gravity (such as around a black hole), the effects are quite small. But they are enough that they can be measured and do have effects on everyday life.
So let's talk details.
Special Relativity
Let's take a look at the equation that deals with time contraction while traveling at high speed:
This is Einstein's Special Theory of Relativity, where
For fun, let's ask the question What would it take for an astronaut to not age? Or in other words, what would it take for an observer on Earth to look at an astronaut and observe he is not aging? The answer is the astronaut would have to be traveling at the speed of light:
So no matter the value of
But perhaps an astronaut will age slower in space than on Earth. Let's explore that  first let's start with the equation again:
According to "the internet", the ISS (International Space Station) travels at around
So in a year of traveling on the ISS, an astronaut would be observed to have aged roughly
General Relativity
Einstein also developed General Relativity, which predicts that someone in a gravitational field will age slower than one outside of it. Since our astronaut will be in a lesser gravitational field than the observer on Earth, the astronaut would be observed to be aging faster! But as we can see from the above, the effects are very very slight.
Putting it together
On Earth, we experience time in our "normal" Earthbased way. As we increase in altitude away from the Earth and the gravity we experience decreases, an observer would see us aging faster. Also, as we orbit the Earth (or head off for the Moon or elsewhere) and our velocity increases, an Earthbased observer would see us aging slower. And it turns out there is a point in space where the two effects cancel (thanks for the tremendous work by @phillipe!). He writes:
The mathematics goes like this. For a satellite in orbit the time dilation due to orbital speed is:
The gravitational time dilation for a distance of
If we need to know at what height the slowing due to speed cancels out the speeding up due to weakened gravity we can compare the
Using Newtonian gravity:
Then:
Divide by
This gives:
So, time passes slower if the satellite, like the ISS, is orbiting lower than an altitude of a half Earth radius, about 3,000 km, and it passes faster if higher, like GPS.
GPS satellites
According to http://metaresearch.org/cosmology/gpsrelativity.asp, "For GPS satellites, GR predicts that the atomic clocks at GPS orbital altitudes will tick faster by about 45,900 ns/day because they are in a weaker gravitational field than atomic clocks on Earth's surface. Special Relativity (SR) predicts that atomic clocks moving at GPS orbital speeds will tick slower by about 7,200 ns/day than stationary ground clocks."
(For some reason, the metaresearch link appears to be broken, but this link from the Astronomy Dept. at Ohio State University provides the same analysis and further discusses how it impacts the GPS network: http://www.astronomy.ohiostate.edu/~pogge/Ast162/Unit5/gps.html)
So the net result is that clocks on GPS satellites, clocks will appear to run faster by 38,700 nanoseconds per day, which is .00000387 seconds, or about 122 seconds per year  so about 2 minutes/year.
ISS
The International Space Station, orbits at 400km which is lower than the 3000 km mark and so time for those astronauts passes slower for them than it does on Earth.
Space Missions
But as speeds increase for astronauts heading to the Moon and elsewhere, Special Relativity causes more time dilation. And as gravity decreases, time speeds up... and so the calculations over which prevails will change...