We study Lambert series generating functions associated with arithmetic functions \(f\), defined by
$$L_f(q)=\sum_{n\ge1}\frac{f(n)q^n}{1-q^n}=\sum_{m\ge1}(f*1)(m)q^m.$$
These expansions naturally generate divisor sums through Dirichlet convolution with the constant-one function and provide a useful framework for enumerating ordinary generating functions of many multiplicative functions in number theory. This paper presents an overview of key properties of Lambert series, together with combinatorial generalizations and a compendium of formulas for important special cases. The emphasis is on formal and structural aspects of the sequences generated by these series rather than on analytic questions of convergence. In addition to serving as an introduction, the paper provides a consolidated reference for classical identities, recent connections with partition-generating functions, and other useful Lambert series expansions arising in applications.