When you invest your money into a fund, where does it initially (literally) go?

Can someone help me understand the basic mechanics/logistics of investing in a private equity fund, hedge fund, index fund, ETF, mutual fund, or other fund? I want to understand just to have a grasp of the industry, not because I am currently looking to invest in a fund with my own money. Specifically, my question is below.

Where does your money literally, physically (well, digitally) go once you invest in a fund? Do you write a check, or send a wire transfer, from your checking account to the fund's checking account, for the fund managers to pool from when they make investments?