Have you ever thought about what kind of job you can get after your math degree? I've compiled a list of the jobs I've seen so far in my own search. I've divided this post into two sections: academia, most of which require a PhD, and industry, which usually does not require a PhD.
The research postdoc has a light teaching load that allows you to do research. This is the typical next research academic position you'll get. Sadly, you might have to go through several of these before you can settle down.
The teaching postdoc requires you to teach around four or five courses a year. Good if you want experience for a teaching career, but it won't allow much time for research.
The tenure-track research professorship allows you to do significant research, forever. These are coveted positions and around twenty will appear that are somewhat close to your area, if you're lucky. Expect 300-800 other people to apply for each position. The best way to get noticed in the truly staggering number of applicants is to have actual strong contacts in departments in which you are applying. Therefore, you need to develop these contacts early on as a PhD student and continue in postdocs.
NB: Some places like Australia don't have the actual tenure concept, but a permanent position there is still pretty secure.
NB: If you're really intent on pursuing the Pure Math PhD->Postdoc(s)->Tenure-track Research career run, the most important thing is to get entrenched with a strong existing research area. That means becoming good contacts with the people in that area. So if you start a PhD in something you really like and your advisor can introduce you to some influential people in the field, and you continue to refine these contacts throughout your postdoc, you'll have a small but nonzero chance. Getting external funding helps as well.
With a tenure-track liberal arts job you'll teach smaller classes with the latest technology and a teaching philosophy while being a proponent of diversity quotas and social justice. Getting this type of job requires a lot of demonstrable teaching qualifications, including experience in teaching and designing actual courses.
The Christian university position can only be found in United States. Job postings emphasise "evangelical Christian mission" and "faith-learning integration".
The one-year position is academia's "gig-economy" job, which exists mainly because there are so many applicants, and thus universities take advantage of the folk that don't get longer-term positions. These can be okay if they are locationally convenient. This is not a job for which you'll want to move to a different country.
The European position has strange promotional stage names like W1. Often you can still teach in English, and you'll benefit from strange European luxuries like health care, fine food, good public transportation, and a reasonable teaching load. Permanent European positions are very attractive.
The exotic position is in a place like China or the Middle East and boasts a radically different way of life than the west, but is nevertheless alluring because of the rapid economic expansion in these areas.
The editor position sometimes comes up on MathJobs. It's like a rare easter egg. It involves doing something like managing MathReviews.
Very few of these actually require a PhD, but a PhD in mathematics can be helpful just for the experience or additional knowledge. Personally, I have found the biggest difference between academic job postings and non-academic job postings is that the latter frequently require experience in specialized technologies that PhDs typically don't use.
The programming job involves programming production-grade systems in languages like C++, Python, and Java. These will often require experience with version control, unit testing, and agile development. Programming jobs are not that difficult to get, and making significant code contributions to or even doing sole development of an open-source project is a great way to get prior experience if you're coming from academia.
NB: Programming jobs are quite varied. Some will use a fair amount of mathematics, but most will require very little.
The finance job centers around trading derivatives (not the calculus kind) using concepts from probability, statistics, game theory, and other econometric-like fields. Interviews involve tricky probability problems, so make sure you refresh your probability and stats knowledge. Finance jobs are good for people who like to solve problems under pressure and who have a strong interest in economics and markets.
NB: There are several kinds of finance-related jobs. Some are mainly live-trading. Others involve various proportions of programming and research, and don't include any front-line trading.
The data analysis job is a hugely popular offering these days. Just search 'math' on a job site and expect to find hundreds of these posted every week. These often require not only knowledge but also experience in statistics and stats software like R. Additional common requirements include big-data experience in Hadoop, experience in turning data into practical recommendations, and machine learning experience. So while you can gain much of the requirements as someone transitioning from pure math, stats PhDs who actually tend to get practical experience will undoubtedly have advantages over pure math people.
The modeling job isn't very common. It involves modeling of real-world phenomenon with tools other than statistics and data science. Examples of some of these companies are ThinkTank Maths in the UK and Palantir, which sometimes also has positions specifically for mathematicians.
The non-mathematical job covers anything that has nothing to do with math, like teaching English or joining the Olympic curling team. Such a job is worth considering if you have other skills and the lifestyle of academia or technical industry isn't for you.