Depending on how much and how heavy you have few options:
This is the fastest and easiest way, but it's also most expensive. Pricing is per weight, but there is also limit on volume. Minimum charged weight is 45kg (100lbs) which means minimum charge of about €100, subsequent charges actually drop. Here is example from US Airways Cargo:
Minimum weight: 45 kg.
Minimum airfreight charges: € 97.00
Up to 45 kg: € 6.26 per kg.
45 - 100 kg: € 4.75 per kg.
100 - 250 kg: € 3.35 per kg.
250 - 500 kg: € 2.86 per kg.
Over 500 kg: € 2.59 per kg.
This is much slower, for example to west coast US it will take 5-7 weeks just sailing, not to mention waiting for available ship. On the other hand it's cheaper for heavy things, especially if you choose whole container.
Typically it's priced per volume, rather than weight. For smaller stuff there is option of sending boxes, priced per volume in increments of 1 cubic meter (example price: €300 1st m³ and €190 for each next m³). For relocating a lot of stuff there is and option of renting whole containers, which come in two sizes, 20ft (33m³) and 40ft (66m³). Prices on containers vary, but generally if you need to move more than 15m³, you're better off with whole container.
I've checked Dutch company Worldwide Baggage Services for information, I imagine there ought to be similar companies elsewhere in Europe.