Motivated by biological imaging problems such as cryo-electron microscopy, multi-reference alignment (MRA) studies the recovery of a latent signal from noisy observations corrupted by unknown transformations. In this talk, I focus on recent work that formulates MRA directly in function space, removing restrictive discretization and bandlimited assumptions and revealing a new connection between functional MRA and classical deconvolution with replicated measurements. Leveraging a multivariate extension of Kotlarski’s identity, this approach enables explicit recovery of non-bandlimited signals from second-order statistics and yields sample-complexity guarantees that depend on interpretable parameters such as signal smoothness, noise variance, and correlation length scale. I will place these results in the context of earlier work on functional MRA under random dilations, where Fourier invariants can be systematically unbiased to accommodate non-compact group actions. Finally, I will outline ongoing work on functional multi-target detection, which extends MRA to regimes where multiple copies of a signal appear at unknown locations within a large, noisy observation.