This thesis deals with several subjects from counting complexity and arithmetic circuit complexity.The first part explores the complexity of counting solutions to conjunctive queries, which are a basic class of queries from database theory. We introduce a parameter, called the quantified star size of a query phi, which measures how the free variables are spread in phi. As usual in database theory, we associate a hypergraph to a query phi. We show that for classes of queries for which these associated hypergraphs have bounded generalized hypertree width, bounded quantified star size exactly characterizes the subclasses of queries for which counting the number of solutions is tractable. In the case of bounded arity, this allows us to fully characterize the classes of conjunctive queries for which counting the solutions is tractable. Finally, we also analyze the complexity of computing the quantified star size of a conjunctive query.In the second part we characterize different classes from arithmetic circuit complexity by different means, including conjunctive queries and constraint satisfaction problems, graph polynomials on bounded treewidth graphs, and an extension of the classical arithmetic branching program model by stack memory.In particular, this yields new characterizations of the arithmetic circuit class VP, a class that is central to the area but arguably not well understood.Finally, the third part studies the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of its monomials. We show that these problems are complete for different levels of the counting hierarchy, which had few or no known natural complete problems before.