This paper involves generalizing the Goldblatt-Thomason and the Lindström characterization theorems to first-order modal logic.1 bounded morphic images, generated subframes and disjoint unions and reflects ultrafilter extensions. Our first objective here is to provide a formulation of this theorem in FML.The second aim of this paper is to study Lindström type theorems for first-order modal logic. These theorems determine the maximal expressive power of logics in terms of model theoretic concepts. In a seminal paper [15], Lindström proved that any abstract logic extending first-order logic with compactness and the Löwenheim-Skolem property is not more expressive than the first-order logic.To formulate our problems in technical terms, it is worth it to have a quick overview of basic notions of FML. Suppose τ is a language consisting of countable numbers of countably many constant and relational symbols. The usual syntactical conventions of first-order logic are assumed here. In particular, the notions of τ -terms and τ -atomic formulas are defined in the usual way. Furthermore, first-order modal τ -formulas are defined inductively in the usual pattern as follows:where P is an n-ary predicate and t 1 , . . . , t n are τ -terms. Bound and free variables in a formula are defined as in first-order logic. A τ -sentence is a formula without any free variable. The other logical connectives (∨, →, ↔, ∀), and the modal operator ◻ have their standard definitions. A constant domain Kripke model is a quadruple M = (W, R, D, I), where W ≠ ∅ is a set of possible worlds, R ⊆ W × W is an accessibility relation, and D ≠ ∅ is a domain. Moreover, i. for each w ∈ W and any n-ary predicate symbol P , I(w, P ) ⊆ D n , ii. for any w, w ′ ∈ W and each constant c, I(w, c) = I(w ′ , c) ∈ D. More generally, a varying domain Kripke model is a tuple M = (W, R, D, I, {D(w)} w∈W ), where (W, R, D, I) is a constant domain model and for each w ∈ W , D(w) ≠ ∅ is a domain of w such that D = ⋃ w∈W D(w). The pair F = (W, R) and the triple S = (W, R, D) are respectively called frame and skeleton. A Kripke model M = (W, R, D, I) is said to be based on the frame F = (W, R) or skeleton S = (W, R, D). For a given first-order Kripke model M and any w ∈ W , the pair (M, w) is called a pointed model.An assignment is a function σ, which assigns to each variable v an element σ(v) inside D. Two assignments σ and σ ′ are x-variant if σ(y) = σ ′ (y) for all variables y ≠ x. We denote this by σ ∼ x σ ′ .The notation t M,σ is used for the interpretation of t in M under assignment σ.