Microscopy is an essential tool for life sciences. Thanks to the development of confocal and multiphoton microscopy, scientists are able to obtain high-resolution 3D views of biological specimens. Nevertheless, spatial variations in the index of refraction within specimens cause aberrations that degrade the quality of the 3D views. One can tackle this issue by implementing adaptive optics (AO) techniques, whereby an active element such as a deformable mirror (DM) is used to suppress the aberrations. In this thesis we consider the problem of estimating aberrations in microscopy. Well-established methods to measure aberrations, such as Shack-Hartmann wavefront sensing, cannot be easily applied due to the lack of well-defined reference wavefronts within specimens. Instead, one can consider wavefront sensorless AO (WFSless-AO), where aberrations are estimated indirectly using a suitable image quality metric. In practice, a series of trial aberration corrections are applied with the DM until the image quality metric is maximised. One can reduce the number of necessary trial corrections by modelling the image quality metric, so that the overall image acquisition time is minimised, and side effects such as photobleaching and phototoxicity are curtailed. Quadratic polynomials have been used extensively to model image quality metrics in microscopy. In the first part of this thesis, the problem of computing the parameters of the polynomial directly from input-output measurements is solved using a mathematical optimisation. Once the parameters are known, the aberration estimation problem is formulated into a linear least-squares optimisation, which requires a minimum of N + 1 trial corrections to estimate N orthogonal aberration modes, such as Zernike polynomials. Both the computation of the parameters and the aberration estimation are validated experimentally using an optical breadboard. In the second part of this thesis, we implement a WFSless-AO algorithm in a second-harmonic microscope. To achieve a more refined aberration correction, we compute the least-squares estimate of the aberration by solving a non-convex optimisation problem. Aberration correction experiments are performed using a biologically relevant specimen. In the last part of this thesis, we consider using a phase retrieval algorithm to correct aberrations. We propose an algorithm that uses three measurements of the point-spread function of the optical system. The phase retrieval problem is formulated using the extended Nijboer-Zernike theory, and it is solved using PhaseLift, a signal recovery method based on convex optimisation. The feasibility of this approach is demonstrated by performing aberration correction experiments using an optical breadboard.