X-ray crystallography is a form of microscopy that allows the three-dimensional arrangement of atoms belonging to molecules within crystals to be determined. In this method, a crystal is illuminated with a beam of X-rays and the diffracted amplitudes resulting from the illumination are measured and computationally processed to enable the electron density of the unit molecule, or the unit cell, constituting the crystal to be calculated. The recent development of the X-ray free-electron laser (XFEL) provides new routes for determining molecular structures via its ability to generate intense but brief X-ray pulses. These new instruments enable diffraction measurements to be obtained from crystals that have a small number of unit cells, referred to as nanocrystals, and molecular structure determination via this technique is known as serial femtosecond nanocrystallography (SFX). This thesis is concerned with the characterisation of diffraction data obtained from SFX experiments and the techniques for reconstructing the electron density of the molecule from such data. The noise characteristics of diffraction measurements from nanocrystals is developed. Methods for directly inverting nanocrystal diffraction to obtain the electron density of the molecule are analysed and an approach to ameliorate the effect of noise is proposed and evaluated by simulation. A model for diffraction by nanocrystals that include the effects of different unit cell arrangements and incomplete unit cells on the crystal surface is also developed and explored by simulation. The diffraction by finite crystals is shown to be equal to the incoherent average over a set of unit cells that contain different molecular arrangements related to the symmetry of the crystal at hand. The problem of image reconstruction under this circumstance is investigated. The more general problem of reconstructing multiple, unrelated, objects from their averaged diffraction is also explored and uniqueness properties along with reconstruction algorithms developed. The problem of reconstructing multiple, related, unit cells is studied and preliminary results are obtained. These results show that iterative phase retrieval algorithms can in principle be adapted to reconstruct the electron density of a crystalline specimen from the data obtained in SFX and the retrieval of phases from the diffracted intensity averaged over multiple objects is feasible.