In this thesis, the aim is to present some new classes of non–static and static, spherically symmetric solutions of the Einstein–Maxwell field equations representing compact objects with negative pressure. Throughoutthisthesisthespace–timegeometryisspherical,theradial pressure is negative, and the matter density equals the negative value of the radial pressure (either it is considered or it comes out as a consequence of the calculations). Several non–static solutions are found by taking an ansatz for the components of the metric tensor and on thesquareofelectricfieldintensity. Thesolutionsareshowntosatisfy physical boundary conditions associated with the exact solutions of the Einstein–Maxwell field equations. Due to negative pressure, these solutions can model physical systems such as expanding compact objects containing negative pressure. Petrov and Segr´e classifications that these obtained solutions admit are also discussed in detail. Two staticsolutionsofthefieldequationsarealsoobtainedwiththeansatz similar to that for the non–static cases in order to have a look how the solutions behave for these kind of ansatz in static geometry. All the physicalconditionsareshowntobesatisfiedforthestaticsolutionsand itisshownthatthesesolutionsdescribecompactobjectswithnegative pressure.