Dwarf galaxies are related to important cosmological questions, and central to our understanding of the physics of galaxy formation. In this thesis, I present the results of cosmological, hydrodynamical simulations of the formation and evolution of dwarf galaxies. I compare the simulation results with observations, and interpret them in the context of a Lambda-CDM cosmology. In high resolution simulations of isolated dwarf galaxies, I show that a combination of supernova feedback and the cosmic UV background results in the formation of galaxies with properties similar to the Local Group dwarf spheroidals, and that both effects are strongly moderated by the depth of the gravitational potential. The simulations naturally reproduce the observed scaling relations between luminosity and mass-to-light ratio, and between total stellar mass and metallicities. The final objects have halo masses between 2.3 x 10^8 and 1.1 x 10^9 solar masses, mean velocity dispersions between 6.5 and 9.7 kms^-1, stellar masses ranging from 5 x 10^5 to 1.2 x 10^7 solar masses, median metallicities between [Fe/H]=-1.8 and -1.1, and half-light radii of the order of 200 to 300 pc, all comparable with Local Group dwarf spheroidals. The simulations also indicate that the dwarf spheroidal galaxies observed today lie near a mass threshold around 10^9 solar masses, in agreement with stellar kinematic data, where supernova feedback not only suffices to completely expel the interstellar medium and leave the residual gas-free, but where the combination of feedback, UV radiation and self-shielding establishes a dichotomy of age distributions similar to that observed in the Milky Way and M31 satellites. A second line of work has been the analysis of the dwarf galaxy population resulting from the Aquila simulation. By simultaneously including the formation of a Milky Way type galaxy along with ~500 dwarf-sized haloes in the mass range of ~10^8 - 10^10 solar masses, this simulation allows a study of the effect of the environment on dwarf galaxy evolution. I study the relative importance, and interplay, of the different mechanisms for gas loss, and compare the properties of the satellites with those of isolated dwarf galaxies. A third set of simulations focuses on the formation of dwarf galaxies in a representative sample of haloes extracted from the Millennium-II simulation. The six haloes in these simulations all have a z=0 mass of ~10^10 solar masses and show different mass assembly histories, which are reflected in different star formation histories. The galaxies reach final stellar masses in the range of 5 x 10^7 - 10^8 solar masses, consistent with other published simulations of galaxy formation in similar mass haloes. The resulting objects have structures and stellar populations consistent with dwarf elliptical and dwarf irregular galaxies. However, in a Lambda-CDM universe, 10^10 solar mass haloes must typically contain galaxies with much lower stellar mass than these simulations predict, if they are to match observed galaxy abundances. The dwarf galaxies formed in my own and all other current hydrodynamical simulations are more than an order of magnitude more luminous than expected for haloes of this mass. I discuss the significance and possible implications of this result for cosmological models, and for the assumptions about the physics of galaxy formation. Finally, I present preliminary results of a direct comparison between hydrodynamical simulations and semi-analytical models for the formation of dwarf galaxies. Current semi-analytical models, which are tuned to match the statistical properties of galaxies, do not agree with the predictions of hydrodynamical simulations for individual objects. Conversely, when tuned to accurately reproduce the simulations, semi-analytical models can give a more qualitative interpretation of the simulation results, in terms of equations of galaxy formation. The combination of the two methods allows an extrapolation from individual cases to cosmological volumes, not currently attainable with direct simulations alone.