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The aim of seismic imaging is to produce maps indicative of spatial variations in properties of the Earth's subsurface. To create such images geophysicists use seismograms of energy measured over time by receivers at fixed observation points. These seismograms partially sample the seismic wavefield and are used to estimate the interactions between the seismic wavefield and the subsurface heterogeneity. However, because observation points are limited spatially the true interactions are unknown so approximations must be made to estimate these interactions. Conventional methods make the assumption that seismic waves observed in the seismograms reflect or diffract at most once from the subsurface heterogeneity (the so-called Born approximation). This assumption allows a low-wave-number smoothly varying estimate of the subsurface velocity structure to be used to back extrapolate the observed seismic wavefield to points inside the subsurface { producing an estimate of the subsurface seismic wavefield. However, this approximation can lead to inaccuracies when the seismograms contain energy that has reflected or diffracted more than once. In this thesis we create a suite of methods that offer a solution to this problem in a variety of scenarios. The majority of this thesis focuses on a set of novel techniques called Marchenko methods. These enable us to project seismograms to points in the subsurface - creating seismograms as though they had been measured at each subsurface point - while accounting for many of the complex, multiply-reflected seismic wave interactions that take place in the real Earth's subsurface. As a result images created using Marchenko imaging contain a reduction in artifacts that usually contaminate subsurface images due to multiply-reflected seismic waves. This thesis has four main aims which are addressed in consecutive chapters: (1) To introduce Marchenko methods with the minimum amount of mathematics required to understand how the methods iterate to a solution, and to provide a well-commented, easily editable MATLAB code package for demonstration and training purposes. (2) The second aim is to understand the application of Marchenko methods in a three-dimensional world, that is to say we investigate the implications of three-dimensional data, subsurface structures, wavefields and acquisition geometries on the results of Marchenko redatuming and imaging. (3) In a third set of results we aim to incorporate the additional wavefield sampling of vertical seismic profile (VSP) data (measured in boreholes in the Earth's subsurface) into Marchenko imaging with the emphasis being on improving imaging of vertical and near vertical subsurface interfaces. (4) The aim of the final set of results is to use multiply scattered (particularly duplex waves) as well as primary (singly scattered) waves to image the subsurface, again to improve imaging of vertical or near vertical interfaces but this time only using surface seismic data. Overall the results of this thesis demonstrate the effectiveness of Marchenko methods to redatum and image accurately when only low-wavenumber smoothly varying estimates of the subsurface velocity structure are available. We demonstrate the applicability of the methods to three-dimensional problems and a means to include VSP data into the method. Finally, we also redefine the conditions used to create subsurface images, allowing us to image using singly and multiply scattered seismic waves.