The price of the deal, set to be finalised later this year, was not disclosed.
The businesses, based in the UK, Italy and Dubai, primarily serve ultra-high net worth and high net worth clients across Europe, the bank said.
The acquired businesses will be integrated into Credit Suisse's private banking and wealth management division.
The Swiss bank said the transaction would complement its wealth management business in Europe, notably in the UK market where the acquisition will put the bank among the top 10 wealth managers.
"Accelerating our growth momentum in our international markets and in our UHNW (ultra-high net worth) client segment remains a key priority for Credit Suisse," said Romeo Lacher, Credit Suisse's head of private banking for western Europe.