Automated planning problems classically involve finding a sequence of actions that transform an initial state to some state satisfying a conjunctive set of goals with no temporal constraints. But in many real-world problems, the best plan may involve satisfying only a subset of goals or missing defined goal deadlines. For example, this may be required when goals are logically conflicting, or when there are time or cost constraints such that achieving all goals on time may be too expensive. In this case, goals and deadlines must be declared as soft. I call these partial satisfaction planning (PSP) problems. In this work, I focus on particular types of PSP problems, where goals are given a quantitative value based on whether (or when) they are achieved. The objective is to find a plan with the best quality. A first challenge is in finding adequate goal representations that capture common types of goal achievement rewards and costs. One popular representation is to give a single reward on each goal of a planning problem. I further expand on this approach by allowing users to directly introduce utility dependencies, providing for changes of goal achievement reward directly based on the goals a plan achieves. After, I introduce time-dependent goal costs, where a plan incurs penalty if it will achieve a goal past a specified deadline. To solve PSP problems with goal utility dependencies, I look at using state-of-the-art methodologies currently employed for classical planning problems involving heuristic search. In doing so, one faces the challenge of simultaneously determining the best set of goals and plan to achieve them. This is complicated by utility dependencies defined by a user and cost dependencies within the plan. To address this, I introduce a set of heuristics based on combinations using relaxed plans and integer programming formulations. Further, I explore an approach to improve search through learning techniques by using automatically generated state features to find new states from which to search. Finally, the investigation into handling time-dependent goal costs leads us to an improved search technique derived from observations based on solving discretized approximations of cost functions.