We present nested sampling for factor graphs (NSFG), a novel nested sampling approach to approximate inference for posterior distributions expressed over factor-graphs. Performing such inference is a key step in simultaneous localization and mapping (SLAM). Although the Gaussian approximation often works well, in other more challenging SLAM situations, the posterior distribution is non-Gaussian and cannot be explicitly represented with standard distributions. Our technique applies to settings where the posterior distribution is substantially non-Gaussian (e.g., multi-modal) and thus needs a more expressive representation. NSFG exploits nested sampling methods to directly sample the posterior to represent the distribution without parametric density models. While nested sampling methods are known for their powerful capability in sampling multi-modal distributions, the application of the methods to SLAM factor graphs is not straightforward. NSFG leverages the structure of factor graphs to construct informative prior distributions which are efficiently sampled and provide notable computational benefits for nested sampling methods. We present simulated experiments which demonstrate that NSFG is more robust and computes solutions over an order of magnitude faster than state-of-the-art sampling techniques. Similarly, we compare NSFG to state-of-the-art Gaussian and non-Gaussian SLAM approaches and demonstrate that NSFG is notably more robust in describing non-Gaussian posteriors.