A regular paving (RP) is a finite succession of bisections that partitions a multidimensional box into sub-boxes using a binary tree-based data structure, with the restriction that an existing sub-box in the partition may only be bisected on its first widest side. Mapping a real value to each element of the partition gives a real-mapped regular paving (RMRP) that can be used to represent a piecewise-constant function density estimate on a multidimensional domain. The RP structure allows real arithmetic to be extended to density estimates represented as RMRPs. Other operations such as computing marginal and conditional functions can also be carried out very efficiently by exploiting these arithmetical properties and the binary tree structure. The purpose of this thesis is to explore the potential for density estimation using RPs. The thesis is structured in three parts. The first part formalises the operational properties of RP-structured density estimates. The next part considers methods for creating a suitable RP partition for an RMRP-structured density estimate. The advantages and disadvantages of a Markov chain Monte Carlo algorithm, already developed, are investigated and this is extended to include a semi-automatic method for heuristic diagnosis of convergence of the chain. An alternative method is also proposed that uses an RMRP to approximate a kernel density estimate. RMRP density estimates are not differentiable and have slower convergence rates than good multivariate kernel density estimators. The advantages of an RMRP density estimate relate to its operational properties. The final part of this thesis describes a new approach to Bayesian inference for complex models with intractable likelihood functions that exploits these operational properties.