In this talk, I present several methods for adding by forcing a non-saturated Aronszajn tree, and describe some consistency results obtained by such forcings. The first adds almost disjoint subtrees (or alternatively almost disjoint automorphisms) of a free Suslin tree, while preserving CH. With this forcing we obtain a model with no Kurepa trees but which has an almost Kurepa Suslin tree, solving open problems due to Jin-Shelah, Moore, and Bilaniuk. The second adds an almost Kurepa Suslin tree with a non-saturated square by ccc forcing using the $\varrho$-function of Todorcevic over a model satisfying the $\square$-principle. Finally, assuming the failure of Chang's conjecture, there is a ccc forcing adding a strongly non-saturated Aronszajn tree defined using a weak version of a rho-function, which provides another solution to a problem of Moore but in which CH fails. This talk includes joint work with Martinez Mendoza and Stejskalova.