Whatever their historic, architectural or cultural differences, one thing unites New York and London, two of the world’s most celebrated cities – their theatrelands. Although they’re nearly 3,500 miles apart, New York’s Broadway and London’s West End jointly epitomise the very finest in on-stage spectacles the world has to offer – whether they are musicals, mysteries, comedies or classic dramas.

The chance to see some of the world’s finest actors live on stage is one that few visitors to either city can resist. Unlike most other popular entertainments – cinema, TV or even mammoth sporting events – chances to see musicals and plays are few and far between, with tickets to the hottest show in town sometimes sold out months – or even years – in advance.

For those in the know, however, seats for even the most in-demand performances of the season – and prime ones at that – can always be found. All it takes is a little resourcefulness. And a fair amount of cash.

First of all, before you begin foraging for tickets to that must-see matinee or sold-out Saturday, you have to know what’s really worth watching. Naturally, this changes from month to month and year to year. At present, though, it’s hard to imagine that anything – on either side of the Atlantic – could match the popularity and critical acclaim of Hamilton, the smash-hit musical currently playing to packed houses in Broadway’s Richard Rodgers Theatre.

An extraordinary and unlikely piece, it’s actually an 18th century historical drama set to the rhythms and rhymes of contemporary New York hip-hop. It tells the true(ish) tale of Alexander Hamilton, an American revolutionary and politician who rose from a poor background to hold high office, before being brought down by feuding and sexual skulduggery.

Few would have thought that a musical based on long forgotten political factionalism and raps about monetary policy could ever be more than a minor cult hit. Hamilton’s sheer exuberance, humour and inventive wordplay, however, have seen it win overwhelming admiration from audiences and critics alike.

If you like the sound of it, but you’re heading for London rather than New York, don’t despair – just postpone your trip a little. It’s due to open in the West End in the autumn of next year.

Already playing to sell-out audiences in both cities is The Book of Mormon, a musical comedy from the creators of South Park, arguably the world’s most outrageous cartoon series. A hilarious take-down of religion, The Book of Mormon is filthy, funny and flippantly offensive. And yet – somehow – sweetly innocent too.

Since it opened five years ago, it has scooped up several armfuls of prestigious theatre awards – including nine Tonys, four Laurence Oliviers and a Grammy.

Altogether more conventional is Beautiful – a musical taking its inspiration from the life and works of Carole King, the celebrated American singer-songwriter. The show has proved something of a surprise hit in both London and New York.

The production uses many of King’s most famous songs to tell the story of her romantic and professional relationship with Gerry Goffin, her writing partner and, now, ex-husband. It’s uplifting and powerful, and its success even led to King coming out of semi-retirement to play a one-off gig in London’s Hyde Park.

If you’re looking for something a little more contemporary, however, there are two shows that debuted on Broadway this year that dazzled New York audiences and critics alike. Waitress is a feel-good feminist-lite tale of life in a Deep South diner, while The Humans is a bitter-sweet study of American family life. While it’s a thought-provoking drama, it’s probably not one for those who like to leave the theatre humming a memorable ditty or two.

Looking further ahead, Robert de Niro – that most legendary of movie stars – is set to make his Broadway debut next month. This will see him treading the boards in A Bronx Tale, a musical take on his own 1993 mob drama. De Niro’s involvement alone pretty much guarantees a sell-out run.

This aside, 2017 looks set to be a big year for Broadway revivals, with new productions of The Glass Menagerie – the classic drama that first made Tennessee Williams’ name – and Hello Dolly, the evergreen musical that sees Bette Midler take on the title role.

Another show destined to be a 2017 hit states-side has already wowed audiences in London. Groundhog Day – a new musical based on Bill Murray’s ever-popular 1993 big screen comedy of the same name – famously tells the comic tale of Phil Connors, a TV weatherman of no fixed moral compass, who is condemned to live the same day over and over again until he mends his ways. It has received a series of rave reviews since opening at London’s Old Vic theatre in August and it opens on Broadway in March.

The hottest show currently playing in the West End, though, is undoubtedly Harry Potter and The Cursed Child. This – the stage debut of the bespectacled English wizard – has been hailed as a truly magical masterpiece. It’s a sequel to the much-loved book and films and features a middle-aged Harry, startling special effects and a thrilling time-travelling plot.

Perhaps the only downside is its length. In total, it stretches over five hours and is split into two separate parts. This can either be seen on the same day or over two separate nights. None of this, though, seems to have deterred the famously fanatical Harry Potter fanbase, with tickets for the show possibly the hardest to secure on the planet. Pretty much every seat is sold out until April next year – and even then, all that seems to be available is a single seat in the stalls for Part 2. Probably with an obscured view.

Of course, getting seats – especially good ones – for any of the more popular productions can be something of a drama in its own right. Many shows are apparently sold out months in advance or the only tickets available are for matinees or for seats immediately behind a Doric column.

The truly intrepid ticket hunter, though, can pretty much always find what they’re looking for. The first stop should always be the online secondary ticketing sites, the more reputable of which charge just a small mark-up on the original box office price. Some even offer tickets at discounted prices. But probably only for Phantom of the Opera.

For the most comprehensive range of sold-out Broadway shows, try www.broadwaybox.com. As an alternative, there’s also www.theatermania.com and www.playbill.com, both of which offer discounted tickets to users who join their free online clubs.

For the West End, www.lovetheatre.com is your best bet, while www.lastminute.com is excellent when it comes to securing tickets at short notice. Be warned, though, tickets for London shows are far less likely to be offered at a discount than those in New York. And none of them have Harry Potter tickets at any price. Believe us. We’ve checked. Twice.

Alternatively, you can arrange your theatre tickets beforehand, along with your flights and hotel bookings. Try www.theatrebreaks.com/new-york to search for a combination of a Broadway show and a hotel. You’ll have to call them directly, though, if you want flights included in the package. For those London-bound, www.hoteldirect.co.uk/theatrebreaks.htm offers the widest range of West End combinations.

Those already abroad and without access to the internet – should such creatures exist – might prefer to search for tickets in person. London and New York operate official one-stop ticket booths – both called TKTS but, confusingly enough, both entirely separate organisations – in the centre of their respective theatrelands. Both cater for those searching for last-minute, hard-to-acquire tickets.

The TKTS booth in New York’s Times Square is so big and so busy it’s more of a shop than a stand. If you don’t mind a bit of hustle and bustle and some queuing, it is possible to pick up excellent tickets for pretty much any show you want – many of them at discounts of up to 50 percent.

London’s TKTS in Leicester Square is somewhat smaller, but still sells tickets for almost every West End show (Harry Potter and the Cursed Child being the exception). Their seats tend not to be quite as good as those from its Broadway equivalent. Most, though, are sold at a discount and you can not only buy on the day of the performance, but also for up to a week ahead.

Why go to all the trouble of getting the tickets yourself, though? Many hotels will do it all for you. If your hotel boasts a concierge, simply ask them if they can acquire the tickets you’re after. Almost inevitably, they’ll be able to oblige.

Asked for the inside track, “Daniel” – who works on the concierge desk of one luxury central London hotel – said: “If our guests tell us in the morning what show they would like to see that evening, we can purchase the tickets and have them printed and ready for them that afternoon. It’s very rare that we can’t find what they’ve asked for.”

Josh, a former Manhattan concierge, told a similar tale, saying: “The concierge has special powers that you don’t have. He deals with ticket brokers, while you, a mere mortal, only have access to the box office or illegal and untrustworthy scalpers (ticket touts). Our tickets will be pricey, but they’ll be the only way you’ll get into Hamilton tonight.”

Talking of pricey, some concierges in New York – though reportedly not at the more up-market establishments – will add on a 10-15 percent surcharge on top of the cost of the tickets, while all will expect to be tipped handsomely for their efforts. In top London hotels, however, procuring theatre tickets seems to be part of the service.

Can they find those elusive Harry Potter and the Cursed Child tickets, though? Sadly, it seems no. When asked, Daniel smiles ruefully and shakes his head. It seems some tasks are beyond even the finest concierge.