DNA is a double stranded helical molecule with an intrinsic right handed twist. Its structure can be changed by applying forces and torques in single molecule experiments. In these experiments DNA has been seen to form super-helical structures (supercoils), collapse into tightly condensed states (toroids) and undergo structural changes (phase transitions). Our work focuses on studying all these phenomena by accounting for DNA elasticity, entropic effects due to thermal fluctuations and electrostatics. First, we study the DNA compaction problem in super-helices and toroidal structures. To do so we combine a fluctuating elastic rod model of DNA with electrostatic models for DNA-DNA interactions. Our models are able to predict the onset of the transition to supercoils and toroids under a wide range of experimental conditions. Next, we address DNA phase changes in the presence of mechanical loads.A phenomenon well known from experiments is the overstretching transition associated with the sudden change of DNA extension at high tensions. Depending on the ionic concentration, temperature and pulling rate, DNA can either transform into a melted state (inner strand separation) or S-DNA. Motivated by this, we study the equilibrium and kinetics of the DNA overstretching transitions making use of a quartic potential and non-gaussian integrals to evaluate the free energy of the system. We find that the cooperativity of the transition is a key variable that characterizes the overstretched state. In a separate study we make use of a heterogeneous fluctuating rod model to examine the hypothesis that a newly discovered left-handed form called L-DNA is a mixture of two relatively well-characterized DNA phases - S-DNA and Z-DNA. L-DNA is stable at high tensions and negative twist. We show that if the idea of a mixed state is correct, then the content of S-DNA and Z-DNA varies as a function of the ionic concentration. Finally, we also use our fluctuating rod model to study the mechanical properties of drug-DNA complexes. We show that our methods can predict the results of experiments from various labs if we use only one set of experiments to fit the data to our model.