In this thesis we study some Questions on vector bundles over hypersurfaces. More precisely, for hypersurfaces of dimension $\geq 2$, 
 we study the extension problem of Vector bundles. We try to find some conditions under which a vector bundle over an ample divisor of
 non-singular projective variety, extends as a vector bundle to an open set containing that ample divisor. Our method is to follow the 
 general Groethendieck-Lefschetz theory by trying to show that a vector bundle extension exists over the various thickenings of the ample 
 divisor. For the case of line bundles, our method unifies and recovers the  Noether-Lefschetz theorems by Joshi and Ravindra-Srinivas.
 For rank $>$ 1, we find 2 separate cohomological conditions on bundle which shows the extension over an open set.  For Line bundles, we 
 give a generalized version of Noether-Lefschetz theorem. In the last part of the thesis, we make a specific study of vector bundles over 
 elliptic curves.