tl;dr: No! Arrow functions and function declarations / expressions are not equivalent and cannot be replaced blindly.
If the function you want to replace does not use this, arguments and is not called with new, then yes.

As so often: it depends. Arrow functions have different behavior than function declarations / expressions, so let's have a look at the differences first:

1. Lexical this and arguments

Arrow functions don't have their own this or arguments binding. Instead, those identifiers are resolved in the lexical scope like any other variable. That means that inside an arrow function, this and arguments refer to the values of this and arguments in the environment the arrow function is defined in (i.e. "outside" the arrow function):

// Example using a function expression
function createObject() {
  console.log('Inside `createObject`:', this.foo);
  return {
    foo: 42,
    bar: function() {
      console.log('Inside `bar`:', this.foo);
    },
  };
}

createObject.call({foo: 21}).bar(); // override `this` inside createObject

// Example using a arrow function
function createObject() {
  console.log('Inside `createObject`:', this.foo);
  return {
    foo: 42,
    bar: () => console.log('Inside `bar`:', this.foo),
  };
}

createObject.call({foo: 21}).bar(); // override `this` inside createObject

In the function expression case, this refers to the object that was created inside the createObject. In the arrow function case, this refers to this of createObject itself.

This makes arrow functions useful if you need to access the this of the current environment:

// currently common pattern
var that = this;
getData(function(data) {
  that.data = data;
});

// better alternative with arrow functions
getData(data => {
  this.data = data;
});

Note that this also means that is not possible to set an arrow function's this with .bind or .call.

If you are not very familiar with this, consider reading

• MDN - this
• YDKJS - this & Object prototypes

2. Arrow functions cannot be called with new

ES2015 distinguishes between functions that are callable and functions that are constructable. If a function is constructable, it can be called with new, i.e. new User(). If a function is callable, it can be called without new (i.e. normal function call).

Functions created through function declarations / expressions are both constructable and callable.
Arrow functions (and methods) are only callable. class constructors are only constructable.

If you are trying to call a non-callable function or to construct a non-constructable function, you will get a runtime error.

Knowing this, we can state the following.

Replaceable:

• Functions that don't use this or arguments.
• Functions that are used with .bind(this)

Not replaceable:

• Constructor functions
• Function / methods added to a prototype (because they usually use this)
• Variadic functions (if they use arguments (see below))
• Generator functions, which require the function* notation

Lets have a closer look at this using your examples:

Constructor function

This won't work because arrow functions cannot be called with new. Keep using a function declaration / expression or use class.

Prototype methods

Most likely not, because prototype methods usually use this to access the instance. If they don't use this, then you can replace it. However, if you primarily care for concise syntax, use class with its concise method syntax:

class User {
  constructor(name) {
    this.name = name;
  }
  
  getName() {
    return this.name;
  }
}

Object methods

Similarly for methods in an object literal. If the method wants to reference the object itself via this, keep using function expressions, or use the new method syntax:

const obj = {
  getName() {
    // ...
  },
};

Callbacks

It depends. You should definitely replace it if you are aliasing the outer this or are using .bind(this):

// old
setTimeout(function() {
  // ...
}.bind(this), 500);

// new
setTimeout(() => {
  // ...
}, 500);

But: If the code which calls the callback explicitly sets this to a specific value, as is often the case with event handlers, especially with jQuery, and the callback uses this (or arguments), you cannot use an arrow function!

Variadic functions

Since arrow functions don't have their own arguments, you cannot simply replace them with an arrow function. However, ES2015 introduces an alternative to using arguments: the rest parameter.

// old
function sum() {
  let args = [].slice.call(arguments);
  // ...
}

// new
const sum = (...args) => {
  // ...
};

Related question:

• When should I use arrow functions in ECMAScript 6?
• Do ES6 arrow functions have their own arguments or not?
• What are the differences (if any) between ES6 arrow functions and functions bound with Function.prototype.bind?
• How to use arrow functions (public class fields) as class methods?

Further resources:

• MDN - Arrow functions
• YDKJS - Arrow functions

No.

There are function declarations, function expressions, and arrow functions (the syntax which creates them also being an expression).

(There are also method declarations which can use arrow functions.)

ludwig = () => {
  console.log("Hi from arrow function declaration");
};

This is assigning a value (an arrow function) to a variable.

The variable declaration is missing, so either it appeared earlier or this creates an implicit global (which is forbidden in strict mode).