Under ECMAScript 5, you can combine Object.keys() and Array.prototype.forEach():

var obj = {
  first: "John",
  last: "Doe"
};

//
//	Visit non-inherited enumerable keys
//
Object.keys(obj).forEach(function(key) {

  console.log(key, obj[key]);

});

In ES6/2015 you can loop through an object like this (using the arrow function):

Object.keys(myObj).forEach(key => {
  console.log(key);        // the name of the current key.
  console.log(myObj[key]); // the value of the current key.
});

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In ES7/2016 you can use Object.entries instead of Object.keys and loop through an object like this:

Object.entries(myObj).forEach(([key, val]) => {
  console.log(key); // the name of the current key.
  console.log(val); // the value of the current key.
});

The above would also work as a one-liner:

Object.entries(myObj).forEach(([key, val]) => console.log(key, val));

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In case you want to loop through nested objects as well, you can use a recursive function (ES6):

const loopNestedObj = obj => {
  Object.keys(obj).forEach(key => {
    if (obj[key] && typeof obj[key] === "object") loopNestedObj(obj[key]); // recurse.
    else console.log(key, obj[key]); // or do something with key and val.
  });
};

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The same as function above, but with ES7 Object.entries() instead of Object.keys():

const loopNestedObj = obj => {
  Object.entries(obj).forEach(([key, val]) => {
    if (val && typeof val === "object") loopNestedObj(val); // recurse.
    else console.log(key, val); // or do something with key and val.
  });
};

Here we loop through nested objects change values and return a new object in one go using Object.entries() combined with Object.fromEntries() (ES10/2019):

const loopNestedObj = obj =>
  Object.fromEntries(
    Object.entries(obj).map(([key, val]) => {
      if (val && typeof val === "object") [key, loopNestedObj(val)]; // recurse
      else [key, updateMyVal(val)]; // or do something with key and val.
    })
  );

Another way of looping through objects is by using for ... in and for ... of. See vdegenne's nicely written answer.

The problem with this

for (var key in validation_messages) {
   var obj = validation_messages[key];
   for (var prop in obj) {
      alert(prop + " = " + obj[prop]);
   }
}

is that you’ll also loop through the primitive object's prototype.

With this one you will avoid it:

for (var key in validation_messages) {
   if (validation_messages.hasOwnProperty(key)) {
      var obj = validation_messages[key];
      for (var prop in obj) {
         if (obj.hasOwnProperty(prop)) {
            alert(prop + " = " + obj[prop]);
         }
      }
   }
}

Using Underscore.js’s _.each:

_.each(validation_messages, function(value, key){
    _.each(value, function(value, key){
        console.log(value);
    });
});

If you use recursion you can return object properties of any depth-

function lookdeep(object){
    var collection= [], index= 0, next, item;
    for(item in object){
        if(object.hasOwnProperty(item)){
            next= object[item];
            if(typeof next== 'object' && next!= null){
                collection[index++]= item +
                ':{ '+ lookdeep(next).join(', ')+'}';
            }
            else collection[index++]= [item+':'+String(next)];
        }
    }
    return collection;
}

//example

var O={
    a:1, b:2, c:{
        c1:3, c2:4, c3:{
            t:true, f:false
        }
    },
    d:11
};
var lookdeepSample= 'O={'+ lookdeep(O).join(',\n')+'}';


/*  returned value: (String)
O={
    a:1, 
    b:2, 
    c:{
        c1:3, c2:4, c3:{
            t:true, f:false
        }
    },
    d:11
}

*/

This answer is an aggregate of the solutions that were provided in this post with some performance feedbacks. I think there is two use cases and the OP didn't mention if he needs to access the keys in order use them during the loop process.

I. The keys need to be accessed

✔ The of and Object.keys approach

let k;
for (k of Object.keys(obj)) {

    /*        k : key
     *   obj[k] : value
     */
}

✔ The in approach

let k;
for (k in obj) {

    /*        k : key
     *   obj[k] : value
     */
}

Use this one with caution, as it could print prototype'd properties of obj

✔ The ES7 approach

for (const [key, value] of Object.entries(obj)) {

}

However, at the time of the edit I wouldn't recommend the ES7 method, because JavaScript initializes a lot of variables internally to build this procedure (see the feedbacks for proof). Unless you are not developing a huge application which deserves optimization, then it is OK, but if optimization is your priority, you should think about it.