Note: Most of the answers cover function pointers which is one possibility to achieve "callback" logic in C++, but as of today not the most favourable one I think.

What are callbacks(?) and why to use them(!)

A callback is a callable (see further down) accepted by a class or function, used to customize the current logic depending on that callback.

One reason to use callbacks is to write generic code which is independant from the logic in the called function and can be reused with different callbacks.

Many functions of the standard algorithms library <algorithm> use callbacks. For example the for_each algorithm applies an unary callback to every item in a range of iterators:

template<class InputIt, class UnaryFunction>
UnaryFunction for_each(InputIt first, InputIt last, UnaryFunction f)
{
  for (; first != last; ++first) {
    f(*first);
  }
  return f;
}

which can be used to first increment and then print a vector by passing appropriate callables for example:

std::vector<double> v{ 1.0, 2.2, 4.0, 5.5, 7.2 };
double r = 4.0;
std::for_each(v.begin(), v.end(), [&](double & v) { v += r; });
std::for_each(v.begin(), v.end(), [](double v) { std::cout << v << " "; });

which prints

5 6.2 8 9.5 11.2

Another application of callbacks is the notification of callers of certain events which enables a certain amount of static / compile time flexibility.

Personally, I use a local optimization library that uses two different callbacks:

• The first callback is called if a function value and the gradient based on a vector of input values is required (logic callback: function value determination / gradient derivation).
• The second callback is called once for each algorithm step and receives certain information about the convergence of the algorithm (notification callback).

Thus, the library designer is not in charge of deciding what happens with the information that is given to the programmer via the notification callback and he needn't worry about how to actually determine function values because they're provided by the logic callback. Getting those things right is a task due to the library user and keeps the library slim and more generic.

Furthermore, callbacks can enable dynamic runtime behaviour.

Imagine some kind of game engine class which has a function that is fired, each time the users presses a button on his keyboard and a set of functions that control your game behaviour. With callbacks you can (re)decide at runtime which action will be taken.

void player_jump();
void player_crouch();

class game_core
{
    std::array<void(*)(), total_num_keys> actions;
    // 
    void key_pressed(unsigned key_id)
    {
        if(actions[key_id]) actions[key_id]();
    }
    // update keybind from menu
    void update_keybind(unsigned key_id, void(*new_action)())
    {
        actions[key_id] = new_action;
    }
};

Here the function key_pressed uses the callbacks stored in actions to obtain the desired behaviour when a certain key is pressed. If the player chooses to change the button for jumping, the engine can call

game_core_instance.update_keybind(newly_selected_key, &player_jump);

and thus change the behaviour of a call to key_pressed (which the calls player_jump) once this button is pressed the next time ingame.

What are callables in C++(11)?

See C++ concepts: Callable on cppreference for a more formal description.

Callback functionality can be realized in several ways in C++(11) since several different things turn out to be callable*:

• Function pointers (including pointers to member functions)
• std::function objects
• Lambda expressions
• Bind expressions
• Function objects (classes with overloaded function call operator operator())

_{* Note: Pointer to data members are callable as well but no function is called at all.}

Several important ways to write callbacks in detail

• X.1 "Writing" a callback in this post means the syntax to declare and name the callback type.
• X.2 "Calling" a callback refers to the syntax to call those objects.
• X.3 "Using" a callback means the syntax when passing arguments to a function using a callback.

Note: As of C++17, a call like f(...) can be written as std::invoke(f, ...) which also handles the pointer to member case.

1. Function pointers

A function pointer is the 'simplest' (in terms of generality; in terms of readability arguably the worst) type a callback can have.

Let's have a simple function foo:

int foo (int x) { return 2+x; }

1.1 Writing a function pointer / type notation

A function pointer type has the notation

return_type (*)(parameter_type_1, parameter_type_2, parameter_type_3)
// i.e. a pointer to foo has the type:
int (*)(int)

where a named function pointer type will look like

return_type (* name) (parameter_type_1, parameter_type_2, parameter_type_3)

// i.e. f_int_t is a type: function pointer taking one int argument, returning int
typedef int (*f_int_t) (int); 

// foo_p is a pointer to function taking int returning int
// initialized by pointer to function foo taking int returning int
int (* foo_p)(int) = &foo; 
// can alternatively be written as 
f_int_t foo_p = &foo;

The using declaration gives us the option to make things a little bit more readable, since the typedef for f_int_t can also be written as:

using f_int_t = int(*)(int);

Where (at least for me) it is clearer that f_int_t is the new type alias and recognition of the function pointer type is also easier

And a declaration of a function using a callback of function pointer type will be:

// foobar having a callback argument named moo of type 
// pointer to function returning int taking int as its argument
int foobar (int x, int (*moo)(int));
// if f_int is the function pointer typedef from above we can also write foobar as:
int foobar (int x, f_int_t moo);

1.2 Callback call notation

The call notation follows the simple function call syntax:

int foobar (int x, int (*moo)(int))
{
    return x + moo(x); // function pointer moo called using argument x
}
// analog
int foobar (int x, f_int_t moo)
{
    return x + moo(x); // function pointer moo called using argument x
}

1.3 Callback use notation and compatible types

A callback function taking a function pointer can be called using function pointers.

Using a function that takes a function pointer callback is rather simple:

 int a = 5;
 int b = foobar(a, foo); // call foobar with pointer to foo as callback
 // can also be
 int b = foobar(a, &foo); // call foobar with pointer to foo as callback

1.4 Example

A function ca be written that doesn't rely on how the callback works:

void tranform_every_int(int * v, unsigned n, int (*fp)(int))
{
  for (unsigned i = 0; i < n; ++i)
  {
    v[i] = fp(v[i]);
  }
}

where possible callbacks could be

int double_int(int x) { return 2*x; }
int square_int(int x) { return x*x; }

used like

int a[5] = {1, 2, 3, 4, 5};
tranform_every_int(&a[0], 5, double_int);
// now a == {2, 4, 6, 8, 10};
tranform_every_int(&a[0], 5, square_int);
// now a == {4, 16, 36, 64, 100};