I am going to review the status of loop quantum gravity as it is seen from the point of view of a Lorentz covariant approach to loop quantization. I'll start from a brief review of the standard loop approach based on the SU(2) gauge group, its main results and problems. Then I'll present another approach which is based on a canonical formulation of general relativity which is explicitly covariant under the local Lorentz transformations. It allows to overcome several problems of the standard loop quantization and at the same time shows that the latter breaks the diffeomorphism invariance and is not a correct way for quantizing gravity. In the covariant framework I'll derive a new spectrum of the area operator, both for spacelike and timelike surfaces, and show that its predictions agree with the spin foam quantization.

I present results on non-perturbative effects in the c=1 string theory. First, I describe a geometric picture found in the CFT framework which gives an interpretation of D-branes in non-critical strings in terms of a complex curve associated with any closed string background. I show that its c=1 limit is degenerate and the degeneracy can be removed by considering a condensation of tachyon modes. Using the matrix model description, I calculate the leading as well as the subleading non-perturbative corrections to the string partition function. We find them by using the Toda integrable structure and from the realization of 2D string theory in terms of free fermions. Both methods give the same result which is also interpreted through correlation functions of a bosonic field. The leading corrections can be interpreted in terms of localized D-branes, whereas the sub-leading ones do not have a simple D-brane description.