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The central focus of this thesis is the study of DNA, a biological molecule of paramount significance. It stores all genetic information in a series of base pairs, the four-letter biological equivalent of the alphabet, and is involved in a plethora of functions within the cell. We are interested in its physical properties, and particularly in three distinct aspects: micromechanics, capture by nanopore and hybridization. The first topic deals with the mechanical properties of a single molecule. Similar to cords from daily life, DNA can be stretched, bent and twisted. In fact, it is hardly ever free of mechanical stress within the cell, as it is constantly deformed by binding proteins. Unlike cords, however, DNA has a special feature: it twists upon bending and bends upon twisting. The first part of the thesis investigates how this twist-bend coupling influences the mechanics of DNA. The way DNA is forced to pass through a nanometer-sized hole is the focus of the second topic. Such a nanopore can be an open cylindrical protein, embedded in a cell membrane. Since DNA is negatively charged, it can be "squeezed" through the pore by applying a voltage difference between the two sides of the membrane. We study the capture from the one side of the membrane into the nanopore, and find that DNA first needs to overcome an entropic barrier. The third topic is devoted to DNA hybridization, the formation of the familiar double helix from two separate strands. The probability of this depends on how good the match is between the two sets of base pairs, which is exploited by several techniques to identify a sequence. A central problem is the detection of a low-abundance sequence (e.g. tumor DNA) in a sea of other sequences (e.g. healthy DNA), often differing by a single base pair. Here, we present a novel strategy that offers a greatly-enhanced detection sensitivity, with a direct diagnostic application. Throughout this work, we rely on theory (read: pen-and-paper calculations) and simulation (read: experiments on a computer) to make predictions about biophysical phenomena. In the end, we test their validity and relevance by comparing to experiments performed by collaborating groups. status: published