Wetting of solid structures is ubiquitous in energy, health, and manufacturing applications, such as coating and painting processes, lab-on-a-chip, and additive manufacturing. Detailed experiments and simulations enhance our understanding and enable the design and control of improved systems and processes. These wetting processes are greatly influenced by the behavior of the contact line between the involved fluids and solid substrates. Here, I will use the Cahn-Hilliard-Navier-Stokes (CHNS) equations with Generalized Navier boundary conditions to accurately predict the two-phase flow including contact line slip and dynamic contact angles. Then I will discuss multiple stable discretization schemes and present simulation results for droplets, bubbles and thin liquid films. Build upon the detailed simulations, mathematical optimization with partial differential equations as constraints can greatly accelerate the usual engineering design process and improve the control of complex liquid-solid interactions. I will derive the adjoint for the CHNS model above and discuss the implementation. Then I will utilize this for the efficient, gradient-based calculation of optimal trajectories for droplet control, where the discrete contact angle between the liquid drop and solid substrate serves as the time-varying control parameter. Increasingly, we aim to directly integrate noisy and incomplete measurements into complex simulations to infer system properties. This often involves formulating and approximating inverse problems with large-scale partial differential equations, forming the basis for uncertainty quantification, data-driven modeling, scientific machine learning, and data assimilation. I will introduce a Bayesian framework for the approximation of such inverse problems for wetting processes via CHNS. Then I will replace the intractable likelihood function with stochastic surrogates based on transport maps in a simulation-based inference scheme and apply it to estimate fluid properties from images of drops impinging on solid substrates.