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Given a fractal set E on the plane and a set F of directions, can we find one direction L in F such that the orthogonal projection of E onto L is large? Suppose f is a function whose Fourier transform is supported on a curved manifold (for instance, a sphere),  what can we say about this function? It turns out that these two questions are related. We will survey some classical and recent projection theorems and discuss their relation to Fourier restriction theory.