Constrained Counting and Sampling are two fundamental problems in Computer Science, where the task is to count the number of solutions or satisfying assignments to a given set of constraints, or to sample a solution uniformly at random. Counting and sampling along with their approximate and weighted variants have been extensively studied in both theory and practice. However, this research effort has been disjointed, resulting in significant gaps in knowledge. On one hand, algorithms with worst-case polynomial running times are considered to be the gold standard by the theory community, but rarely scale well in practice. On the other hand, powerful general-purpose algorithms and tools developed by the AI and Formal Methods communities often fail to scale on ‘easy’ problems with polynomial upper bounds. The goal of this dissertation is to illuminate and address this disconnect. Specifically, we develop flexible techniques that natively exploit the structure inherent in domain-specific constraints. This often leads to significant performance gains over the popular approach which attempts to shoehorn all constraints to fit a rigid algorithm. Motivated by numerous practical applications and a lack of practically scalable tools with strong theoretical guarantees, we present new solutions for the concrete problems of DNF-Counting, conditional counting, computing the matrix permanent, sampling traces of a transition system and weighted sampling from low-treewidth CNF formulas. Our empirical analyses reveal a nuanced picture wherein our approaches are seen to be a valuable addition to an algorithmic portfolio.