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In this thesis, we present two approaches to a rigorous mathematical and algorithmic foundation of quantitative and statistical inference in constraint-based natural language processing. The first approach, called quantitative constraint logic programming, is conceptualized in a clear logical framework, and presents a sound and complete system of quantitative inference for definite clauses annotated with subjective weights. This approach combines a rigorous formal semantics for quantitative inference based on subjective weights with efficient weight-based pruning for constraint-based systems. The second approach, called probabilistic constraint logic programming, introduces a log-linear probability distribution on the proof trees of a constraint logic program and an algorithm for statistical inference of the parameters and properties of such probability models from incomplete, i.e., unparsed data. The possibility of defining arbitrary properties of proof trees as properties of the log-linear probability model and efficiently estimating appropriate parameter values for them permits the probabilistic modeling of arbitrary context-dependencies in constraint logic programs. The usefulness of these ideas is evaluated empirically in a small-scale experiment on finding the correct parses of a constraint-based grammar. In addition, we address the problem of computational intractability of the calculation of expectations in the inference task and present various techniques to approximately solve this task. Moreover, we present an approximate heuristic technique for searching for the most probable analysis in probabilistic constraint logic programs.