Most "open-ended" mutual funds these days allow you to buy or sell an arbitrary number of shares of the mutual fund from the mutual fund itself, which (respectively) creates or destroys those shares and sells them shortly after the market closing that follows the placing of the order; that is, if you place an order on a weekend before a Monday holiday, your transaction takes place late Tuesday afternoon, at the same time as if you place it early Tuesday afternoon. The price at which the transaction takes place is the "Net Asset Value" (NAV); the fund calculates the price of all of the assets it owns, divides it by the number of shares outstanding, and buys or sells the shares pro rata. For large cap stock mutual funds, this works quite smoothly almost always, and it's extraordinary for it to work poorly; the assets have closing auctions on various stock exchanges that tend to be fairly competitive and result in official "closing" prices that are fairly unbiased and accurate predictors of the price at which the asset may next be bought and/or sold (or, indeed, the prices at which they may have been bought or sold earlier that afternoon). These assets have prices to a sufficient extent to make this rule work.
There has been increasing concern lately about other kinds of funds, which own assets that do not have very well-defined prices, and here's an example of a fund whose client doesn't like how it made up the prices, but I tend to think the procedure was reasonable. The client sold a very large amount of shares back to the fund — the fund would have to sell assets (at the very least to get back to its usual level of cash holdings), and to value some of the assets that were particularly hard to sell, asked a few dealers how much it could sell them for. They got, naturally, a lower price than they would have had to pay to buy them, or the price at which they last traded; this resulted in a lower NAV than one of those higher prices would have.
It seems to be conventional to use "last-traded price" in many contexts where that isn't a particularly unbiased predictor of where the asset can be sold in the future; if bonds have dropped in price over the last couple days, and a mutual fund has a fair number of bonds that haven't traded in that time, the "last-traded price" is an overestimate of any meaningful "price" that the bond has now, and fund holders redeeming at the (inflated) NAV will be overpaid, to the disadvantage of continuing fund-holders. A bank — I think it was Barclays; I should have made a note — has indicated that, for the high-yield bond mutual funds it was looking at, this problem could be solved by letting sellers (i.e. people redeeming their mutual fund holdings) choose between a 2% discount in the price or a 30 day delay in settlement, either compensating the fund (i.e. the other, continuing investors in the fund) for the adverse selection problem or waiting until bond prices have updated.
While this is mostly intended as a solution to the adverse selection problem, it also somewhat mitigates the problem I'm noting here; a fund with 30 days' advanced notice can sell assets more carefully than one trying to raise a lot of cash before 4:30 this afternoon.
I kind of think that funds (especially with illiquid assets) should quote bid-offer spreads that reflect the bid-offer spreads of the underlying assets; if all of the assets have bids that are 95% of their offers, the fund-wide spreads might be somewhat tighter than that, reflecting the fund's discretion in choosing which assets to sell and a level of liquidity support (cash holdings or perhaps a short-term line of credit, if that's allowed); they should probably, as with most active exchanges, depend to some degree on traded volume, so that large orders to sell would face larger discounts, and they should probably allow orders to buy to net off against orders to sell before hitting either the bid or the offer. What I'm mostly looking at is essentially a closing auction with the fund filling imbalances; as I may have noted before, open-ended funds with fees for orders that create imbalances and closed-ended funds that make markets in their own stocks kind of converge on each other, and that's what seems like the right approach to me.