In this thesis, a natural generalization of the definition of a Hopf algebroid is introduced, internal to any symmetric monoidal category with coequalizers that commute with the monoidal product. In this thesis we also construct a symmetric monoidal category (indproVect,\(\tilde{\otimes} ,k)\) of filtered-cofiltered vector spaces, whose morphisms are linear maps which in a weak sense respect the filtrations and cofiltrations, and whose monoidal product is the usual tensor product of vector spaces formally completed and with a corresponding filtration of cofiltrations. We prove that this category satisfies the above conditions for the existence of internal Hopf algebroids. It contains two dual subcategories, the category (indVect,\(\otimes ,k)\) of filtered vector spaces and the category (proVect,\(\hat{\otimes} ,k)\) of cofiltered vector spaces. The monoidal product in it combines the ordinary tensor product and a completed tensor product. An important class of Hopf algebroids over a noncommutative base is comprised of smash products of a Hopf algebra \(H\) and a braided-commutative algebra \(A\) in the category of YetterDrinfeld modules over \(H\). Such Hopf algebroids are called scalar extensions. In this thesis, we prove that the smash products in which \(H\) and \(A\) are replaced by their analogues in the monoidal category of filtered-cofiltered vector spaces have the structure of Hopf algebroids in that monoidal category. By doing this, we set the basis for studying the Heisenberg doubles \(A^\ast \sharp A\) in which \(A\) is an infinite-dimensional Hopf algebra instead of a finite-dimensional one, among other examples, and the existence of the Hopf algebroid structure on them internal to the category indproVect. We then study Hopf pairings of a filtered Hopf algebra \(A\) and a cofiltered Hopf algebra \(H\) which are non-degenerate in the variable in \(H\), and find sufficient conditions for \(A\) to be a braided-commutative Yetter-Drinfeld module algebra over \(H\) in the indproVect category. A smaller class of examples is also studied, for which \(A\) is a Hopf algebra countably filtered by finite-dimensional vector spaces. Here we find necessary and sufficient conditions on Hopf algebras \(A\) and \(A^\ast\), or \(A\) and \(H\), in the form of finite dimensionality of the adjoint orbits of \(A\) and the existence of certain canonical elements in \(H \sharp A\). Thus a construction of some filtered-cofiltered Hopf algebroids of scalar extension type is obtained. Important examples of such scalar extensions are the ones with \(A\) the universal enveloping algebra \(U(\mathfrak{g})\) of a finite-dimensional Lie algebra \(\mathfrak{g}\). When \(H\) is equal to its algebraic dual \(U(\mathfrak{g})^\ast\) with induced cofiltration, the corresponding scalar extension, that is the Heisenberg double of \(U(\mathfrak{g})\), can be identified as an algebra with the algebra of differential operators on the formal neighborhood of the unit of a Lie group integrating \(\mathfrak{g}\), suggesting applications in geometry and mathematical physics.