  In this paper, we introduce a class of function spaces called K\"othe-Herz
spaces $E(\mathcal{X})$. These spaces are similar to amalgam spaces and are
characterized by a local component given by a countable family
$\mathcal{X}=\left( X_{\alpha }\right) _{\alpha \in I}$ of quasi-normed
function spaces, and a global component $E$, which is a quasi-normed sequence
space. We investigate various geometric and topological properties inherited by
$E(\mathcal{X})$ from its components, such as their completeness, duality,
order continuity, ideal and Fatou properties, in an abstract setting. In
addition, we provide a Banach function space characterization for
$E(\mathcal{X})$, which allows us to understand its structure and behavior more
deeply. Furthermore, by appropriate amalgamation of Lorentz spaces (Orlicz
spaces) and Lebesgue sequence spaces, we define Lorentz-Herz spaces
(Orlicz-Herz spaces) as a particular case of $E(\mathcal{X})$, which are still
generalizations of the classical Herz spaces. In this context (especially
Lorentz-Herz spaces), we establish previously studied properties, demonstrate
interpolation results, and prove the boundedness of important sublinear
integral operators with kernels that satisfy a size condition.
