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An arrow function in TypeScript is a syntactically compact alternative to a regular function expression, characterized by the => (fat arrow) token and the lexical binding of the this context. Unlike standard functions, arrow functions do not possess their own this, arguments, super, or new.target bindings, and they cannot be invoked with the new keyword as constructors.

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Syntax and Type Annotations

TypeScript extends standard JavaScript arrow functions by allowing explicit type annotations for parameters and the return value. 
const functionName = (param1: string, param2: number): boolean => {
    // execution statements
    return true;
};
Arrow functions support both block bodies (requiring an explicit return statement) and concise bodies (which implicitly return the evaluated expression). 
// Block body with explicit return type
const add = (x: number, y: number): number => {
    return x + y;
};

// Concise body with implicit return
const multiply = (x: number, y: number): number => x * y;

// Concise body returning an object literal (requires parentheses)
const createPoint = (x: number, y: number): { x: number; y: number } => ({ x, y });

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Function Type Expressions

In TypeScript, the arrow syntax is also heavily utilized to declare function signatures (Function Type Expressions) within interfaces, type aliases, or inline parameter types. In this context, the arrow denotes the return type rather than a function implementation. 
// Defining a function type using arrow syntax
type MathOperation = (a: number, b: number) => number;

// Implementing the function (parameter types are inferred contextually)
const subtract: MathOperation = (a, b) => a - b;

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Lexical this Binding and Restrictions

Arrow functions resolve this lexically, capturing the this value of the enclosing execution context at runtime. TypeScript accurately models this standard JavaScript behavior in its type system. Because arrow functions establish this lexically, TypeScript strictly prohibits declaring a this parameter type annotation within an arrow function signature. Attempting to do so results in a compiler error. 
// Compiler Error: An arrow function cannot have a 'this' parameter.
const invalidArrow = (this: Window, event: Event) => {};

// Valid: Standard function expression allows explicit 'this' typing
const validFunction = function(this: Window, event: Event) {};

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Overloading Restrictions

Unlike standard function declarations, arrow functions cannot be directly overloaded using multiple signature declarations immediately preceding the implementation. To achieve overloading with an arrow function, the variable must be explicitly typed using a type alias or interface that defines the overloaded signatures. 
// Standard function overloading (Valid)
function process(x: number): number;
function process(x: string): string;
function process(x: any): any { return x; }

// Arrow function overloading requires a separate type definition
type ProcessOverloads = {
    (x: number): number;
    (x: string): string;
};

// The implementation is assigned to the explicitly typed variable
const processArrow: ProcessOverloads = (x: any): any => x;

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Generic Arrow Functions

Arrow functions can accept generic type parameters. The type parameter declaration <T> is placed immediately before the opening parenthesis of the parameter list. 
const getFirstElement = <T>(arr: T[]): T | undefined => {
    return arr[0];
};
Note on TSX syntax: When writing generic arrow functions in .tsx files (React), the compiler may misinterpret the <T> syntax as a JSX tag. To resolve this, a constraint or a trailing comma must be added to the generic type parameter to disambiguate it. 
// Disambiguating generic arrow functions in .tsx files
const getFirstElementTsx = <T extends unknown>(arr: T[]): T | undefined => arr[0];
// Alternatively using a trailing comma: <T,>

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Rest Parameters and Default Values

Arrow functions fully support TypeScript’s rest parameters and default parameter initializers, with types applied directly to the parameter declarations. 
// Default parameters
const greet = (name: string, greeting: string = "Hello"): string => `${greeting}, ${name}`;

// Rest parameters
const sumAll = (...numbers: number[]): number => {
    return numbers.reduce((acc, curr) => acc + curr, 0);
};
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