Abstract: Consider a Besov space X=X(R^3) which is invariant under the spatial scaling of the Navier-Stokes equations and for which a local existence theory for any initial datum in X has been developed. We show that for any local Navier-Stokes solution u starting from an initial datum u_0 in X and which (hypothetically) has a singularity at time T>0, the (spatial) X norm of u(.,t) must become unbounded as t approaches T. This extends a famous result of Escauriaza-Seregin-Sverak (2003) which establishes the same result for X=L^3(R^3), which is continuously embedded in all of the Besov spaces mentioned above. Our proof uses profile decompositions for bounded sequences in Banach spaces as well as the related "critical element" theory developed for and implemented in the setting of critical dispersive and hyperbolic equations by C. Kenig and F. Merle. The program for Navier-Stokes (the first parabolic setting in which the critical element method was successfully implemented) was started in collaboration with Carlos Kenig (Chicago) and completed in collaboration with Isabelle Gallagher (Paris) and Fabrice Planchon (Nice). As a by-product of the tools developed for this program, we were able to easily extend a recent result of Rusin-Sverak regarding "minimal blow-up data" for Navier-Stokes in critical spaces to all of the spaces mentioned above.