With the introduction of (i|android|windows)phones, things have changed, and to get a correct and complete solution that works on any device is really time-consuming.
You can have a peek at https://realfavicongenerator.net/favicon_compatibility or http://caniuse.com/#search=favicon to get an idea on the best way to get something that works on any device.
You should have a look at
http://realfavicongenerator.net/
to automate a large part of this work, and probably at https://github.com/audreyr/favicon-cheat-sheet to understand how it works (even if this latter resource hasn't been updated in a loooong time).
One complete solution requires to add to you header the following (with the corresponding pictures and files, of course) :
<link rel="shortcut icon" href="favicon.ico">
<link rel="apple-touch-icon" sizes="57x57" href="apple-touch-icon-57x57.png">
<link rel="apple-touch-icon" sizes="114x114" href="apple-touch-icon-114x114.png">
<link rel="apple-touch-icon" sizes="72x72" href="apple-touch-icon-72x72.png">
<link rel="apple-touch-icon" sizes="144x144" href="apple-touch-icon-144x144.png">
<link rel="apple-touch-icon" sizes="60x60" href="apple-touch-icon-60x60.png">
<link rel="apple-touch-icon" sizes="120x120" href="apple-touch-icon-120x120.png">
<link rel="apple-touch-icon" sizes="76x76" href="apple-touch-icon-76x76.png">
<link rel="apple-touch-icon" sizes="152x152" href="apple-touch-icon-152x152.png">
<link rel="apple-touch-icon" sizes="180x180" href="apple-touch-icon-180x180.png">
<link rel="icon" type="image/png" href="favicon-192x192.png" sizes="192x192">
<link rel="icon" type="image/png" href="favicon-160x160.png" sizes="160x160">
<link rel="icon" type="image/png" href="favicon-96x96.png" sizes="96x96">
<link rel="icon" type="image/png" href="favicon-16x16.png" sizes="16x16">
<link rel="icon" type="image/png" href="favicon-32x32.png" sizes="32x32">
<meta name="msapplication-TileColor" content="#ffffff">
<meta name="msapplication-TileImage" content="mstile-144x144.png">
<meta name="msapplication-config" content="browserconfig.xml">
In June 2016, RealFaviconGenerator claimed that the following 5 lines of code were supporting as many devices as the previous 18 lines:
<link rel="apple-touch-icon" sizes="180x180" href="/apple-touch-icon.png">
<link rel="icon" type="image/png" href="/favicon-32x32.png" sizes="32x32">
<link rel="icon" type="image/png" href="/favicon-16x16.png" sizes="16x16">
<link rel="manifest" href="/manifest.json">
<link rel="mask-icon" href="/safari-pinned-tab.svg" color="#5bbad5">
<meta name="theme-color" content="#ffffff">
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