MomentJs is an open source library that provides utility functions to handle date and time. The library works well with Salesforce Locker Service.

Here is an example component showing how to integrate MomentJs into Lightning Components. The steps to integrate the library are as follows: 

1. Create a static resource hosting MomentJs: The source code for the library can be obtained from https://MomentJs.com/downloads/Moment.Js. Download and create a static resource file named MomentJs. 

2. Use the ltng:require tag and leverage the afterScriptsLoaded event to load the JavaScript files and styles. The following is simple component code that demonstrates the use of the ltng:require tag:

<aura:component implements="force:appHostable,flexipage:availableForAllPageTypes,flexipage:availableForRecordHome,force:hasRecordId" access="global" >
<ltng:require scripts="{!$Resource.MomentJs}"
afterScriptsLoaded="{!c.afterScriptsLoaded}" />

3. Use third-party JavaScript functions and code in the afterScriptsLoaded function. The controller file code to test the various functions provided by the library is as shown in the following code snippet:

({
afterScriptsLoaded : function(component, event, helper) {
    console.log(moment().format('MMMM Do YYYY, h:mm:ss a')); // July 11th 2017, 11:23:19 am
    console.log(moment().format('dddd')); // Tuesday
    console.log(moment().format("MMM Do YY")); // Jul 11th 17
    console.log(moment().format('YYYY [escaped] YYYY')); // 2017 escaped 2017
    console.log(moment().format()); // 2017-07-11T11:23:19-04:00
    console.log(moment("20111031", "YYYYMMDD").fromNow()); // 6 years ago
    console.log(moment("20120620", "YYYYMMDD").fromNow()); // 5 years ago
    console.log(moment().startOf('day').fromNow()); // 11 hours ago
    console.log(moment().endOf('day').fromNow()); // in 13 hours
    console.log(moment().startOf('hour').fromNow()); // 27 minutes ago
    console.log(moment().subtract(10, 'days').calendar()); // 07/01/2017
    console.log(moment().subtract(6, 'days').calendar()); // Last Wednesday at 11:28 AM
    console.log(moment().subtract(3, 'days').calendar()); // Last Saturday at 11:28 AM
    console.log(moment().subtract(1, 'days').calendar()); // Yesterday at 11:28 AM
    console.log(moment().calendar()); // Today at 11:28 AM
    console.log(moment().add(1, 'days').calendar()); // Tomorrow at 11:28 AM
    console.log(moment().add(3, 'days').calendar()); // Friday at 11:28 AM
    console.log(moment().add(10, 'days').calendar()); // 07/21/2017
    console.log(moment.locale()); // en
    console.log(moment().format('LT')); // 11:29 AM
    console.log(moment().format('LTS')); // 11:29:23 AM
    console.log(moment().format('L')); // 07/11/2017
    console.log(moment().format('l')); // 7/11/2017
    console.log(moment().format('LL')); // July 11, 2017
    console.log(moment().format('ll')); // Jul 11, 2017
    console.log(moment().format('LLL')); // July 11, 2017 11:29 AM
    console.log(moment().format('lll')); // Jul 11, 2017 11:29 AM
    console.log(moment().format('LLLL')); // Tuesday, July 11, 2017 11:29 AM
    console.log(moment().format('llll'));
    var now = moment();
    console.log(now);
    var day = moment(1318781876406);
     console.log(day);
    }
 })

4. Monitor console.log from the Chrome developer Console to confirm the library is functioning inside Lightning Components. If the library is functioning, the output will resemble the following screenshot: