Once we formalize the syntax of a language, the next step is to formalize its semantics: what programs mean and how they behave. There are three classical approaches. 1) Operational semantics Operational semantics defines meaning by describing how programs execute on an abstract machine. A machine state is typically just a term of the language, and evaluation proceeds by transitions between states. In small step semantics, execution is modeled as a sequence of reduction steps. In big step semantics, evaluation relates a term directly to its final value. Different operational presentations can model different levels of abstraction, and proving their correspondence amounts to proving implementation correctness. Operational semantics answers the question: how does this program run? 2) Denotational semantics Denotational semantics abstracts away from execution and maps programs directly to mathematical objects such as numbers, functions, or more structured semantic domains. Instead of describing steps of evaluation, we define an interpretation function from syntax to a semantic domain. This perspective enables equational reasoning, compositionality, and deep connections with domain theory. Denotational semantics answers the question: what mathematical object does this program denote? 3) Axiomatic semantics Axiomatic semantics shifts the focus from behavior to provable properties. Rather than defining execution and then deriving laws, it takes logical rules as primary. The meaning of a program is given by what can be proven about it. This tradition gave us foundational ideas such as invariants and Hoare logic, emphasizing reasoning as central to programming. Operational semantics models execution. Denotational semantics models mathematical meaning. Axiomatic semantics models provability. For students and engineers working in PL, compilers, or formal verification, understanding these three perspectives is essential. They are not competing views. They are complementary lenses on what programs are and what it means for them to be correct.