Shape constraints on probability distributions, such as monotonicity and log-concavity of the density, arise naturally in statistics. In previous work, these nonparametric estimation problems are typically performed by maximum likelihood estimation. In this talk, we consider estimation of univariate densities using Wasserstein projection, where the shape constraint is captured by a displacement convex set in the Wasserstein space. We discuss properties of the Wasserstein estimator and, through numerical examples, compare it with the maximum likelihood estimator. This is on-going work with Takeru Matsuda.