user924: [FEN ""]

1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.O-O

In recent years, the move 4.O-O has taken over from 4.d3 almost entirely -- in my database of TWIC games, if I look at 2750+ games from the last year, I see 4.0-0 70 times and 4.d3 only 5 times. They mostly seem to transpose into the same positions. This seems to be relatively recent and might be a factor behind the recent popularity of 3.Bc4 (related to the question What happened to the Giuoco Pianissimo? , I turned my comments there into this question).
What's the advantage of 4.O-O over 4.d3?
In case it helps, I seem to recall someone asking this elsewhere (maybe on Reddit?) and a helpful answer linked to some well known commentators explaining it (perhaps Svidler and Gustafsson?). I was at work and didn't watch the video, and I've forgotten where I saw it.

user14202: According to a New in Chess Yearbook 124 article:
"Recently, White has been playing this move order, intended to avoid lines that allow Black the possibility of a quick ...d7-d5(normally, this central break only works when White has played his pawn to c3).
4.c3 Nf6 5.d3 0-0 6.0-0 d5 is precisely what White is trying to avoid...

user11609: I see two reasons why to prefer castling over c3. First reason why I would go for 4.0-0 is that I would like to postpone playing c3 to make lines with d5 riskier. Imagine Marshall Ruy-Lopez, if the pawn was on c2, white would have very easy life, no weakness on d3, Nc3 possibility. And you can always play the c3 later if you like. The second thing may be desire to avoid playing c3 in early phase at all. White players tried to play a4 to create shelter for the bishop on a2, not minding exchange for opponent's bishop, avoiding classical Bb3-c2 plan. 

924: <pre><code>[FEN &quot;&quot;]

1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.O-O
</code></pre>
<p>In recent years, the move 4.O-O has taken over from 4.d3 almost entirely -- in my database of TWIC games, if I look at 2750+ games from the last year, I see 4.0-0 70 times and 4.d3 only 5 times. They mostly seem to transpose into the same positions. This seems to be relatively recent and might be a factor behind the recent popularity of 3.Bc4 (related to the question <a href="https://chess.stackexchange.com/questions/18777/what-happened-to-the-giuoco-pianissimo">What happened to the Giuoco Pianissimo?</a> , I turned my comments there into this question).</p>
<p>What's the advantage of 4.O-O over 4.d3?</p>
<p>In case it helps, I seem to recall someone asking this elsewhere (maybe on Reddit?) and a helpful answer linked to some well known commentators explaining it (perhaps Svidler and Gustafsson?). I was at work and didn't watch the video, and I've forgotten where I saw it.</p>
 14202: <p>According to a New in Chess Yearbook 124 article:
"Recently, White has been playing this move order, intended to avoid lines that allow Black the possibility of a quick ...d7-d5(normally, this central break only works when White has played his pawn to c3).
4.c3 Nf6 5.d3 0-0 6.0-0 d5 is precisely what White is trying to avoid...</p>
 11609: <p>I see two reasons why to prefer castling over c3. First reason why I would go for 4.0-0 is that I would like to postpone playing c3 to make lines with d5 riskier. Imagine Marshall Ruy-Lopez, if the pawn was on c2, white would have very easy life, no weakness on d3, Nc3 possibility. And you can always play the c3 later if you like. The second thing may be desire to avoid playing c3 in early phase at all. White players tried to play a4 to create shelter for the bishop on a2, not minding exchange for opponent's bishop, avoiding classical Bb3-c2 plan. </p>
