We intend to give an elementary talk to explain why (smooth) subvarieties of small codimension are expected to be quite special. We will concentrate on a theorem by Barth stating that a subvariety of dimension $n$ in a projective space of dimension $N$ inherits much of the topology of the projective space, namely the integral cohomology up to order $2n-N$ must be the same. We will give a new geometrical approach to this theorem, which will allow us to extend Barth's theorem to other ambient spaces different from the projective space. We will put all this in relation with the famous Hartshorne's conjecture about subvarieties of small codimension in the projective space