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

Arrow functions => best ES6 feature so far. They are a tremendously
powerful addition to ES6, that I use constantly.

Wait, you can’t use arrow function everywhere in your code, its not going to work in all cases like this where arrow functions are not usable. Without a doubt, the arrow function is a great addition it brings code simplicity.

But you can’t use an arrow function when a dynamic context is required: defining methods, create objects with constructors, get the target from this when handling events.

Arrow functions should NOT be used because:

1. They do not have this

   It uses “lexical scoping” to figure out what the value of “this”
   should be. In simple word lexical scoping it uses “this” from the
   inside the function’s body.

2. They do not have arguments

   Arrow functions don’t have an arguments object. But the same
   functionality can be achieved using rest parameters.

let sum = (...args) => args.reduce((x, y) => x + y, 0);
sum(3, 3, 1) // output: 7

3. They cannot be used with new

   Arrow functions can’t be constructors because they do not have a prototype property.

When to use arrow function and when not:

1. Don’t use to add function as a property in object literal because we
   can not access this.
2. Function expressions are best for object methods. Arrow functions
   are best for callbacks or methods like map, reduce, or forEach.
3. Use function declarations for functions you’d call by name (because
   they’re hoisted).
4. Use arrow functions for callbacks (because they tend to be terser).

They are not always equivalent. Here’s a case where you cannot simply use arrow functions instead of regular functions.

Arrow functions CANNOT be used as constructors

TLDR:

This is because of how Arrow Functions use the this keyword. JS will simply throw an error if it sees an arrow function being invoked as a “constructor”. Use regular functions to fix the error.

Longer explanation:

This is because objects “constructors” rely on the this keyword to be able to be modified.

Generally, the this keyword always references the global object. (In the browser it is the window object).

BUT, when you do something like:

function personCreator(name) {
   this.name = name;
}

const person1 = new personCreator('John');

The new keyword do some of its magic and makes the this keyword that is inside of personCreator to be initially an empty object instead of referencing the global object. After that, a new property called name is created inside that empty this object, and its value will be ‘John’. At the end, the this object is returned.

As we see, the new keyword changed the value of this from referencing the global object to now be an empty object {}.

Arrow functions do not allow their this object to be modified. Their this object is always the one from the scope where they were statically created. This is called Static Lexical Scope. That is why you cannot do operations like bind, apply, or call with arrow functions. Simply, their this is locked to the value of the this of the scope were they were created. This is by design.

And because of this :D, arrow functions cannot be used as “constructors”.

Side Note:

A lexical scope is just the area where a function is created. For example:

function personCreator(name) {
    this.name = name;

    const foo = () => {
        const bar = () => {
            console.log(this); // Output: { name: 'John' }
        }

        console.log(this); // Output: { name: 'John' }
    
        bar();
    }

    foo();
}

const person1 = new personCreator('John');

The lexical scope of bar is everything that is within foo. So, the this value of bar is the one that foo has, which is the one of personCreator.