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This thesis explores different aspects of DNA topology through experimental and numerical techniques. Topology is a vast mathematical field, that deals with the spatial properties of objects undergoing continuous deformations, but here it is restricted to the description of closed space curves such as rings, knots and catenanes. Topology is connected to practical problems in different domains including polymer physics, because a polymer chain is a type of space curve, whose statistical properties are affected by the closure of its ends; and biology, because the double-helical interwinding of the two strands formingDNA is described by topological concepts, and also because a closed DNA molecule can adopt complex topologies like knotted or catenated conformations in vivo. Both physical and biological aspects of topology are investigated in the three chapters of this thesis. Chapter 1 introduces on a general level the experimental techniques, numerical methods and topology concepts that are at the core of the presented research. Chapter 2 is an experimental study of the statistical properties of DNA rings adsorbed on surfaces, and imaged by atomic force microscopy (AFM). DNA rings of different sizes are first studied in a dilute state, allowing us to investigate the consequences of the circular topology on polymer physics of two-dimensional chains. We confirm certain theoretical results, e.g. the topological invariance of the scaling exponent in two dimensions; but we also propose new models taking into account the specificities of the problem, in particular the interplay between excluded volume effects and topology, for the description of standard polymer physics measures like the asphericity or the bond correlation function. In addition to isolated DNA rings, we study DNA rings in a semi-dilute state. Comparing the statistical properties of the DNA rings imaged by AFM with a numerical model, we formulate a simplified description of this system in terms of two dimensional pressurized vesicles. Finally, we study a last case, in which DNA rings are confined within circular boundaries in two dimensions. We show how confinement strongly affects the statistical properties of the DNA chains, and verify that our experimental system is a good model for polymer confinement by comparing the data with numerical simulations. In chapter 3, we use numerical simulations to study two other topological forms taken by closed DNA in vivo: catenanes and knots. In both cases, we analyze the conformations of these objects, in particular as a function of their torsional state, and find complex behaviors proper to these topologies: for example, the large scale supercoiling of toroidal catenanes, or the different response of left- and right-handed knots to torsion. We then focus on the activity of a class of enzymes called topoisomerases, that are able to change the topology of DNA molecules (e.g. by unknotting a DNA knot). In particular, we try to deduce how the structural properties of DNA knots and catenanes affect the activity of one of these enzymes called topo IV, and also simulate its activity. We show that our numerical model provides results that compare well with experimental data, and could therefore provide a basis for the design of new experiments. Finally, we present a new model for the simulation of the activity of another topoisomerase called gyrase, that is responsible for introducing supercoiling into DNA. This model includes a few essential features of the interaction between gyrase and DNA, which permit to efficiently reproduce experimental measurements.