It’s been three and a half years since Apple opened the App Store’s virtual doors, and app mania shows no signs of abating.

The concept of an Apple-managed software store has become so widely embraced that the company even created a Mac version of the App Store in January 2011. But the iOS store isn’t ready to cede the spotlight just yet. A quick glance at the winners of this year’s App Gems honours reveals that developers continue to roll out stellar software that pushes the limits of just what your iPhone, iPod touch, and iPad can do. And with half a million apps now swelling the App Store’s shelves, we expect plenty of good things still to come on the iOS front.

With so many apps to choose from, it’s hard for one person to keep track of the many, many apps that pop up every day, clamouring for attention. Which is why the Macworld team has compiled this definitive list of the greatest, cleverest, most innovative and fun apps we’ve seen in the past twelve months.

Top productivity apps

Our favourite productivity apps each have different ways of helping you get more out of your iOS device. A pair of the winners in this category improve upon the things your iOS device can do already, delivering in ways that Apple’s built-in apps don’t. The others add some new and exciting features to your iPad and iPhone, making you more productive in the process.

iOS 5 introduced a built-in Reminders app to manage tasks, but we prefer Due (£2.99, Phocus, www.dueapp.com) when it comes to quickly keeping track of the many to-dos that come up in a given week. Creating reminders in Due is simple and speedy, but the app’s best feature is its one-tap customisation. You can reschedule reminders, add repeats, recycle old tasks, and email or message others, all without complications or extra hassle. Combine that with a beautiful interface, the ability to sync across devices, and persistent push notifications, and it’s clear why we need no reminder to tout our affection for Due.

iOS devices also include a built-in Calendar app, but, frankly, it leaves a lot to be desired. For an upgrade, turn to Agenda Calendar (69p, App Savvy, www.getappsavvy.com), which uses the same cloud-synced calendar data but makes it easier to view and manage. While Agenda works on both the iPad and the iPhone, it’s the latter version that shines. The app provides five views – multi-month, month, week (a scrollable list), day, and event – and you simply swipe right or left to see more or less detail, respectively. In any view, swipe down or up to look back or forward in your schedule, or tap the date icon to quickly enter a date to view or to search for an event. You can share event details via email, and if you’ve installed the Due app we just mentioned, you can add custom alerts and reminders.

Your iOS device doesn’t come with a way to view Mac or PC screens over a network, but plenty of third-party apps can do so. Few take on the task with as much polish as Screens (£13.99, Edovia, www.edovia.com). The app’s interface is attractive and simple, but underneath that veneer is surprising power. There’s support for multiple authentication methods, SSH tunnelling, and Multi-Touch gestures. And Screens is no slouch when it comes to performance and stability. While it might be pricey compared with most of its competitors, Screens is worth the money to anybody who uses screen sharing on a regular basis.

If you’re one of those ‘power’ iPhone users who likes to send text messages and emails in bulk, Peepo (69p, Vine Creative, www.vinecreative.com) could be a handy tool. The app’s mission is simple: it lets users create groups from the contacts list on the phone so that a group text or email can be sent quickly. So no more adding recipients one by one while composing your message. The app flags people in a circle who don’t have an email address or a text number associated with their contact information in your iPhone, making it less likely that you’ll miss transmitting information to its intended recipients.

Top education apps

The iPad has been making inroads in the classroom ever since its debut. If Apple’s tablet is going to continue on this course in education, it will be with apps like our three favourite educational offerings leading the way.

Toontastic (free, Launchpad Toys, www.launchpadtoys.com) is a creative learning tool aimed at the younger set (ages 4 to 7 for some of the app’s categories, and 8 to 10 for others). It’s obvious when you launch Toontastic that kids will have fun with it, for at least a moment. What’s not so obvious is how they can lose themselves completely in the app and their imaginations – and they’ll be eager to share their creations with you. The app playfully walks young animators through the process of creating full cartoon stories. They pick the setting, the characters, and the soundtrack, and they can record the characters’ dialogue, too. All told, Toontastic is a brilliantly executed app that keeps kids entertained, and fully engaged to boot.

Older students – or anyone with an interest in history – can enjoy an immersive experience of their own with Virtual History Roma (£6.99, Arnoldo Mondadori Editore, www.appixgroup.com). A kind of e-textbook, Virtual History offers a lot of fascinating data on emperors, battles, daily life, and other facets of Roman civilisation. But instead of simply relying on text, the app incorporates images, graphics, and other visual techniques to make that information spring to life. Hold your iPad out and spin around to get a 360-degree view of Imperial Rome. A rotating 3D cutaway gives you an up-close look at the Colosseum. Overlays superimpose the architecture of ancient times on the streets of today. It’s an inventive way of conveying information that illustrates the new ways the iPad can help us learn.

If you are fascinated by the world around you – or rather above you – planetarium software Redshift (£7.99, USM iPhone, www.usmiphone.de) aims to bring the whole of the night sky to your iPad, using your current location to show you precisely which stars, constellations and planets are above you. An observatory option lets you jump from your own location’s view of the sky to any celestial object you’d like to see: the moon, a planet, or a specific star. You can zoom into impressively detailed surface maps or enter an orbit.

One fun feature is 3D Flight mode. Instead of just jumping directly to, for example, Jupiter or Saturn, you can have the app animate the journey past the solar system’s different planets and moons. There’s even the option to travel to stars up to 3,000 light years from earth.