A maximal cofinitary group is a subgroup of S∞, all of whose non-identity elements have only finitely many fixed points and which is maximal with respect to this property under inclusion. The minimal size of a maximal cofinitary group is denoted αg. Another well-known cardinal invariant of the real line, denoted non(M), is the minimum size of a set of reals which is not meager. It is a theorem of ZFC that non(M) ≤ αg. A third invariant of interest for this talk, denoted ∂, is the minimum size of a family Ƒ of functions from ℕ to ℕ which has the property that every function from ℕ to ℕ is eventually dominated by an element of Ƒ. In contrast with αg and non(M), ZFC does not determine any relationship between αg and ∂. Using the method of forcing, one can show that each of the inequalities αg < ∂ and ∂ < αg is relatively consistent with ZFC. The classical forcing techniques seem, however, to be inadequate in addressing the latter inequality. Its consistency was obtained only after a ground-breaking work of Shelah and the appearance of his “template iteration” forcing techniques. Jointly with A. Törnquist we have further developed these techniques to show that αg, as well as some of its relatives, can be of countable cofinality. We define two classes of partial orders and the iteration of arbitrary representatives of these classes along a template, which in particular provides a uniform proof of con(cof(ā) = w) for ā ϵ {α; αp; αg; αe}. We will conclude with the discussion of open questions and directions for further research.