William arrived in an understated navy suit and dark paisley tie while Kate wore a navy and green tartan dress.
The Duchess beamed as she waved at well-wishers on Tuesday.
The royal pair, known as the Earl and Countess of Strathearn when in Scotland, were met by V&A Dundee director Philip Long before entering £80million landmark building for a tour.
The duchess, a patron of the V&A, will make a short speech celebrating the addition of the centre to Scotland and the UK.
V&A Dundee is the first V&A museum anywhere in the world outside London.
It has been designed by Japanese architect Kengo Kuma and sits on the banks of the River Tay as the centrepiece of an ongoing £1 billion regeneration of Dundee's waterfront.
Lord Lieutenant Ian Borthwick, Lord Provost of Dundee, said he was "delighted to welcome the Duke and Duchess of Cambridge to V&A Dundee".