The main actors in this PhD thesis are microtubules, which are dynamic polymers that play many important roles in cells. First, we establish theoretically the effect on ordering of the microtubule nucleations from pre-existing microtubules of the plant cortical array, which is a structure that plays a key role in the plant cell elongation. Following, we develop models that can be very helpful in understanding how the spatial organization of microtubules is influenced by the cell shape. Additionally, we formulate a model that predicts the positioning of the mitotic spindle, which is the cellular structure that segregates the duplicated chromosomes during eukaryotic cell division. Moreover, based on the propensity of microtubules to bind and transport proteins to the cell membrane, we set up a feasible and robust cell polarization mechanism. This may form the basis of a biochemical reconstitution approach to set up polarity in a minimal cell-like environment.