It's 1800, and I have arrived in New York with a check from a bank in Paris. How do I cash it - how does the bank verify it - and how much do I pay for the service?

In A History of the Bank of New York 1784-1884, it's stated that the first American cheques/checks were issued by that bank in around 1784, around 15 years after the formal cheque clearing process was pioneered by the Bank of England. Some British cheques date from the mid-1600s, so the system had been in place a long time and it's not unfathomable to think that some hadn't crept across the Atlantic to the Colonies.

Let's consider what the cheque was at this point - it was in effect a form of exchange note which enabled the writer to write the cheque out, instructing his bank (or the chief cashier) to pay out the funds demanded to the bearer from his account upon presentation of the cheque. Banks would keep the counterfoil from the issued cheque and the cheque itself, and ensure that balance was deducted and paid accordingly. In Britain, the Bank of England had preferred them to banknotes since the mid-1720s, and by 1770 British bank clerks would meet at clearing houses to ensure all the numbers added up. Clerks would receive payment from the clearing house for each cheque from another bank that they had in their possession, having paid out the sum to the bearer already, and that balance would be added to the ledger to be paid in clearing by the bank who issued the cheque, and balanced when the clerk from the concerned bank arrived with payment. They would furthermore show the accounts of cheques issued and cashed within their own bank, although this was less of a concern.

However, with the first American cheques being issued in the mid-1780s, and a cheque written by Thomas Jefferson dating from 1809 being preserved, it's clear that the end of the 18th century was a time of popularisation of the method of payment in the newly-created United States. The first American clearing house wasn't set up until 1853, so efforts to clear cheques were likely slow and informal, meaning that neither banks nor individuals could have total certainty of the surety of a cheque. It's unlikely that the American clearing process was anywhere near as organised as Britain's was at the same point. In 1800, the US silver dollar was only 8 years old and competed as a legal currency against the Spanish dollar and the Mexican silver peso. It's possible that in New York the Dutch lion dollar was still circulating in small amounts - so clerks would be dealing with currencies in possibly four different currencies, which made for a clunky and ineffective system of clearing that required currency conversion, often based on old data.

France was, of course, going through a certain degree of upheaval in 1800 (the Banque de France was reformed from the ruins of the failed Banque Royale in that year) and so it's unclear how willing American financiers would have been to accept French cheques. The Banque de France, in any case, had a monopoly on all finance in France 1800-1815, so cheques would only have been officially redeemable there, and only truly clearable if there were Banque de France representatives in New York or if American banks had bases in Paris through which they could clear their cheques and make sure of the accuracy of their accounts. The newly-introduced franc, based on an ambiguous silver value, was newer than the dollar at this point and had suffered fluctuations in its actual silver value, which would continue until its pegging to gold in 1803.

They may have been redeemable at certain banks for banknotes (the US being, unlike the UK at that point, a big fan of printing and using banknotes), as the bank could control the value of what it was handing out, but in the wake of the gold shortages in the late 1700s they probably weren't redeemable for precious metal, especially not cheques from either an unregulated, de-monopolised private French bank or from the newly-set up Banque de France. In any case, you would have a very hard job finding a bank willing to pay out as there was no effective means of clearing the cheque. Remember that America didn't have a nationwide cheque clearing system until the establishment of the Federal Reserve in 1913. Traveller's cheques were invented in 1772, but don't seem to have been standardised until the late 1800s, meaning that there was no effective international means of cheque clearing apart from branch offices. The first trans-border cheque clearing scheme that didn't require extra efforts only came around in 1969. In the intermittent period, travellers would often use letters of credit from their bank to obtain funds from a bank overseas that would then be chased up, although it was far from a perfect system as it required trust from one bank and guarantees from the other. The difficulty in obtaining funds with a letter of credit was part of the inspiration for the invention of travellers' cheques.

To answer your question, you would have a hard time exchanging the cheque in New York in 1800 unless you: a) were a previous and trusted customer of a bank that was prepared to undergo the lengthy and difficult international clearing process, b) could find a bank willing to exchange it, although security (in precious metal or specie) might have been required, c) had a letter of credit that would allow you to draw on your own account instead. The cheque clearing system still isn't fully international in 2015 - cheques still have to be sent to their parent bank for clearing - and it was much worse in 1800.

From my understanding, for most of the 19th and early 20th century it was the done thing to send money overseas by postal order (from 1881) or money order (from the 1820s onwards) as they were safer than cheques, having been prepaid and therefore unable to bounce.

That said, it appears that banks in London and Paris effectively pegged the pound sterling to the franc from 1800-1812 (source in French), with the Bank of England and Banque de France setting a fixed price of 1 franc = 1 pound, before the era of standardised gold and silver prices in Europe ("bimetalism") from 1821 onwards, so it's possible that French cheques might have been redeemable in pounds at British banks that had offices in New York, presumably at a reduced rate due to the uncertainty of the cheque which would have to be returned to London and cleared with French bankers.