avoir beau (from bonjourfrench.tumblr.com)

I remember being confused about this idiomatic expression when I first encountered it, but it’s actually pretty useful once you understand it.

Avoir beau, followed by an infinitive verb, means ‘however + [verb]’ or ‘whatever + [pronoun/noun] + [verb].’ There’s a sense of doing something in vain to no effect. 

Some examples:

J'ai beau essayer = try as I may
On a beau faire = whatever you do, no matter what one does 
Avoir beau étudier = to study in vain

(source: languageramblings, reposted by bonjourfrench on tumblr)