Given an immersion f of the 2-sphere in a Riemannian manifold (M,g) we study
quadratic curvature functionals of the type \int f(S2) H2,      \int f(S2) |A|2,      \int f(S2)
|Ao|2,
where H is the mean curvature, A is the second fundamental form, and Ao is the
tracefree second fundamental form. Minimizers, and more generally critical points of
such functionals can be seen respectively as GENERALIZED minimal, totally geodesic
and totally umbilical immersions. In the seminar I will review some results
(obtained in collaboration with Kuwert, Riviere and Schygulla) regarding the
existence and the regularity of minimizers of such functionals. An interesting
observation regarding the results obtained with Riviere is that the theory of
Willmore surfaces can be useful to complete the theory of minimal surfaces (in
particular in relation to the existence of canonical smooth representatives in
homotopy classes, a classical program started by Sacks and Uhlenbeck).