What is the MAT?
The Maths Admissions Test (MAT) is the admissions test used by Oxford for degrees in Mathematics. If you’re applying for a Maths or Computer Science degree at Oxford or a Maths degree at Imperial College London, you must sit the MAT just after beginning year 13 in late October/early November. If you apply to study Maths at the University of Warwick then you will be encouraged to take the MAT, though it is not compulsory, as a good performance on the MAT, STEP or TMUA often results in a reduced offer. Other universities can sometimes use MAT in a similar way.
How is the MAT structured?
The MAT lasts 2½ hours and consists of 7 questions. Each candidate answers 5 of these questions. The questions that you are required to answer depends on your degree choice. For instance, Maths candidates must answer questions 15, but Computer Science or Maths with Computer Science students will each have a different selection of questions. Calculators or formula sheets aren’t allowed in the exam.
Question 1, which all students must attempt, is composed of ten multiple choice questions (with five answer options) worth 4 marks each, for a total of 40. The remaining four questions are longer questions and are usually composed of many related parts. Each full question is worth 15 marks each, for a total of 60.
What topics do I need to know?
The MAT syllabus is based on the pure topics in ASlevel Maths, with the addition of a few A2 topics that students are assumed to have covered in September and October of year 13. In particular, students should know about arithmetic and geometric progressions and series. Due to Alevel reform in the UK, the syllabus for the MAT was updated in 2018 and the remainder theorem, radians, and the trapezium rule we removed from the syllabus. Combinations and binomial probabilities, the derivative of e^{kx}, differentiation from first principles, and graphs of log_{a}(x) have been added to the syllabus in their place. Do keep this in mind if you are doing past papers: you may come across questions that you haven’t been taught how to solve, and if you are relying solely on past papers to revise (which isn’t recommended) then you won’t revise any questions on differentiation from first principles, for example.
What is the MAT testing that isn't covered in my ALevels? Why the need for another test?
The goal of the MAT isn’t to test that you know the AS syllabus (though if you don’t know it well, you will probably be in trouble!), it is to see how you respond to questions with a large problemsolving component. That is, a question which is difficult not because of the fact that it explicitly involves complex topics, but because the process of navigating yourself to a solution is complex; it relies on a stroke of inspiration, backed up with a systematic and logical approach in order to solve it. Taking this idea of problem solving vs topic complexity to its extreme, you get questions like the following one from Round 2 of the British Mathematical Olympiad (BMO) in 1994:
“Find the first integer n, which is greater than 1 and is such that the average of 1^{2}, 2^{2}, 3^{2}, ... , n^{2} is itself a square number.”
You could ask an 11yearold this question and they would probably be able to understand it, and yet it appears on perhaps the most difficult Maths test that someone could sit when they are 18 years old. The difficulty arises due to the fact that at the beginning, there seems to be no clear path that will definitely solve the problem (in a reasonable amount of time). A question like this makes one wonder, “What should I do, when I have no idea what to do?!” This is what problem solving is about! The best strategy is often to think of a few ideas and then get stuck into the most appealing one as soon as possible  you will never get anywhere with a blank page. Here are a few strategies that often work, taken from nrich’s guide to problem solving:

Try special cases or a simpler problem.

Work backwards.

Guess and check.

Be systematic.

Work towards subgoals.

Imagine your way through the problem.

Has the plan failed? Know when it’s time to abandon the plan and move on.
In the example above, you might try try special cases e.g. averaging the first few squares to see if you can see a pattern emerging. You could also try to be systematic by writing a general equation which you then try to solve. However, you should keep in mind that at the beginning of a problem like this, you are only experimenting. You are not necessarily hoping to solve the problem from the outset; you only want to gain an insight into how the problem should be solved. As such, you should be willing to abandon a solution that doesn’t look like it will lead anywhere, even if you have invested 5 or 10 minutes into it already. Often, you can still use what you have tried to inform your next attempt.
How should I prepare?
Of course, you should familiarise yourself with the structure and style of the test by sitting a few past papers. (However, do note that the syllabus for the MAT changed in 2018, as mentioned above). Some practice under timed/exam conditions might be useful too, to try to replicate the pressure you will be under in the exam. Since your goal is to train your problem solving skills, perhaps the best thing to do would be to try to find as many different sources of questions with a heavy problemsolving component and practice with those. The more experience you have with different types of problems, the more prepared you will be when you are met with a curveball in the exam.
Some good sources of questions similar to those on the MAT are past papers to other entrance exams. In approximate increasing order of difficulty, these include TMUA, AEA, MAT, STEP I and STEP II. STEP III and the BMO may also be useful. However, STEP III covers the Further Maths syllabus and has a slightly lower problem solving element than STEP II (meaning STEP II is probably better practice) and the BMO doesn’t rely on the ALevel syllabus for the most part (except for a few tricks which you may or may not have been taught) and has a much higher problem solving difficulty than any of the other exams.

The TMUA consists of multiple choice questions and is sat at a similar time to the MAT. This means that you can practice the TMUA to help prepare for the MAT’s multiple choice questions. In addition, if you’re finding the MAT questions too difficult, the TMUA can be a useful stepping stone, since the questions are typically easier than those on the MAT.

The AEA is sat in the Summer alongside your other exams and is similar in structure to standard ALevel papers, though they are significantly harder. Since you would sit this paper later in the year, its syllabus is larger, so if you are using AEA papers to prepare for the MAT, you may need to skip some questions if they aren’t relevant to what you need to know for the MAT. The AEA is also easier than the MAT, so it can be used if you’re finding MAT too hard at the beginning of your revision.

STEP I and II are the entrance exams used by Cambridge for Mathsrelated subjects and these are sat in the Summer alongside your other exams. If you have studied Further Maths, your Cambridge offer would usually involve STEP II and STEP III, otherwise it would involve STEP I. STEP I and II cover the ALevel Maths syllabus, as opposed to STEP III which covers the ALevel Further Maths syllabus. The main difference between STEP I and II is that the problemsolving difficulty in STEP II is much higher than STEP I, which in turn is higher than the difficulty of the MAT. If you are finding that you are scoring well on the MAT past papers (e.g. about 80% or more), you might find STEP I and II to be a good way to push yourself further and prepare yourself for the harder questions on the MAT. Again, you may wish to skip some questions that aren’t on the MAT syllabus.
There are some other great resources that are recommended on Oxford’s website:

Search Underground Mathematics’ bank of MAT questions by topic (go to Browse, Review Questions, Oxford MAT, then put in the topic by changing Line/Station).

Check out nrich’s Advanced Problem Solving Modules for a wellcrafted series of questions to guide you through the process of learning to problem solve in a mathematical context.

Have a look at Cambridge’s STEP support Programme for another guided approach to learning to problem solve. This is much more geared towards STEP, but it will still help in your preparation for the MAT. This page also contains a link to Dr Stephen Siklos’ book on STEP preparation, which is an excellent (and free) resource.
How much preparation should I be doing?
You may be looking at the above list of resources and feeling slightly overwhelmed by the amount of work ahead of you. However, since the MAT is sat in the Autumn term, it is not assumed that you will have spent very much time preparing for it. Of course, the more work you put in, the more likely you will do well. As a rough guide, the absolute minimum would be to do 34 past MAT papers under timed conditions and perhaps a couple TMUA or AEA papers. The best case scenario would be that you’ve completed the first few problem solving modules on nrich, almost all of the MAT past papers and a few of each of the other papers above before you sit the MAT.
Do you have any final tips for the exam?

Guess on the multiple choice questions if you’re stuck. On the multiple choice questions, you either score 4 marks or 0 on each part, so there is no negative marking. If you aren’t sure of an answer but think you might know, you can have a guess and make a mental (or physical) note to go back and explore the question further in the last 1015 mins of the exam.

Be moderately rigorous with proofs. If you are required to prove something in a test, the level of rigour you should apply should be more than you would usually use in an ALevel paper, but less than what would be required in something like STEP or the BMO. If you are given a “prove that” question, you should be somewhat rigorous; ensure that all of your steps are fully explained and don’t make any assertions that you don't prove, unless they are obvious. In “show that” questions, you can be slightly lighter on rigour.

Be sure you aren’t reliant on your formula book or calculator. Calculators or formula sheets aren’t allowed in the exam, so make sure you know the relevant formulae that you might normally look up e.g. for summing a geometric or arithmetic series, or for differentiation by first principles. Additionally, you should memorise the value of the trig functions, since you won’t have a calculator. Also make sure you aren’t relying on a graphical calculator if you would normally use one.
Practice as much as you can! This can’t be emphasised enough. The more practice you have with MATstyle problems, the more prepared you will be for the exam.