I like this word. I don’t know why…perhaps because it is one of those words that seems to express itself in speaking of it. The word means ‘anxiety’—a kind of generalized anxiety with being alive.
The existential philosophers talked a lot about angst. In fact, we normally associate angst with existentialism—existential angst. The word is usually associated with a negative mood such as depression or what Thomas Merton characterized as "the dark night of the soul". I think that Heidegger talked about it as the inherent tension between ‘being’ and ‘non-being’. I think that angst underlies the ‘suffering’ that Buddha associated with human existence and probably is behind the concept of ‘original sin’. Whatever its origins or deeper meanings, it is a day-to-day practical reality for most of us in our unending quest to ‘get it right’ and ‘be happy’.
There are lots of strategies for dealing with angst. There is, of course, self-righteousness. Practitioners of this strategy doggedly attach themselves to some interpretation (or dogma) of "The Truth", usually received directly from God. This makes it possible to give up any other questions and reject other ideas as products of those less enlightened or less blessed with faith. 
Another common strategy is self-medication, usually practiced with some substance or behavior that becomes so all consuming that one doesn’t have the time or energy to pursue deep inquiry into the nature of life. The unfortunate souls who live in denial of their addictions follow a well-trod and predictable path until they ‘hit bottom’ and wake up to the fact that they haven’t been in control of whatever it is that controls them.
A strategy I employed for years was to pursue ‘understanding‘. This strategy keeps one so busy learning and reading and thinking that you hardly have time left for living. I remember doing a self-awareness program in the 70s that proclaimed, "Understanding is the booby prize". I think they were right.
There are others whom I think of as the ‘mood merchants‘, people who become very dramatic and significant about their angst. They are tireless in their dedication to enroll others in tasting every morsel of their despair, every fear and uncertainty, and every self-doubt and unfulfilled desire.
While there are probably innumerable other negative strategies for dealing with angst, there are also several very positive ones.
First and foremost is conversation. In speaking about one’s angst, whether with a trained professional therapist or a good friend, one can illuminate the fact that this is part of the human condition, and that anxiety offers us a pathway to experience serenity. Journaling or writing is another version of this strategy—one that has worked for many famous authors and many unknown, but serene, closet writers.
Many people find that the practice of regular physical activity, either alone or in combination with verbalizing, relieves the stresses associated with anxiety. Except for the weekend athletes and extreme overachievers (who may border on being addicted to this strategy), most dedicate themselves to a balanced variety of healthy disciplines such as yoga or tai chi, dietary regimens and contemplation. Bodywork, including massage and reflexology, and self-induced relaxation techniques like meditation, self-hynopsis and biofeedback, offer additional techniques for dealing with generalized anxiety.
People who have found a larger purpose for their lives discover power and serenity through true service to others.
In the final analysis, the only cure for angst is profound acceptance of what is—choosing life on life’s terms and choosing ourselves to be exactly the way we are. Shakespeare’s adage "To thine own Self be true" pretty well sums it up. When we are authentic, when we don’t resist, when we trust some ‘Higher Power’ or simply the process of life itself, we find serenity and gratitude. And when we discover inner peace, then angst is just a word.