Future radio telescopes promise great advances in resolution and sensitivity. These include the Square Kilometer Array, a two array instrument, in South Africa and Australia. Similarly, the next generation Very Large Array (ngVLA) is being designed for construction in North America. These arrays all promise exceptional advances in sensitivity, angular resolution, and survey speed. The SKA and ngVLA are both specified to have sensitivities at the level of $\mu$Jy's. The SKA-Low instrument will consist of a huge number of dipoles antennas in Australia which is pushing the bounds of current FX correlator technology with $O(n2)$ scaling, where $n$ is the number of antennas. The design proposals for these instruments include a dense core of antennas, necessitating advances in imaging methods for these very dense cores versus more traditionally sparse instruments.

Another ambitious experiment is the Hydrogen Epoch of Reionisation Array (HERA) in South Africa which hopes to make the first direct detection of the Epoch of Reionisation through the red-shifted H{\sc i} signal which is a factor of $105$ smaller than the thermal-like noise.

In this thesis, these problems are tackled by re-examining the underlying principles of interferometry. The first working example of a direct imaging correlator is presented which allows images to be formed directly from the voltages off each antenna in a dense array, without an expensive cross-correlation operation as is typically required. A detailed discussion is given of how standard steps in interferometric imaging differ in this new scheme, including calibration. Additionally the first wide field direct imaging correlator is presented, which allows the problems of non-coplanarity to be dealt with for both sparse and dense arrays in a very efficient manner on modern GPU compute hardware. These are, to the best of the authors knowledge, the only working implementations of a direct imaging correlator for generic arrays with no restrictions on the geometry of the array or homogeneity of constituent receiver elements. These new approaches have been published in the scientific literature as discussed in the Declaration.

Moving on from this, the closure phase bispectrum is presented as a way of uncovering the cosmological Epoch of Reionisation signal from the H{\sc i} line. This is using the HERA telescope, which consists of a dense core of parabolic antennas in a highly redundant layout. A data reduction and processing pipeline for the HERA telescope is constructed and presented, for use with the bispectrum. Initial results towards a cosmologial limit are reported.

The HERA telescope relies on redundancy in its antenna elements for its calibration and measurement strategy. The bispectrum with its unique mathematical propeties, in combination with forward modelling, is shown to be a potent tool for probing departures from the assumed reudundancy. It is shown, through this method, that HERA suffers significant direction-dependent non-redundancies in the dataset used for our analysis, which are extremely difficult to calibrate out.

Finally, the problem of wide-field imaging in next generation arrays is tackled through the development and implementation of a new scheme of wide field imaging. This uses a new method of parallelising the problem of wide-field imaging, and is intended for use with the very large datasets that will be produced by upcoming instruments. Two schemes are introduced: $w$-towers, and Improved $w$-towers. The latter generalises the former in combination with advances in optimal convolution theory for the radio astronomy ``gridding'' problem. The theory behind this approach is explored, and a high performance implementation is presented for $w$-towers and Improved $w$-stacking within Improved $w$-towers.