Early on in my poker career I lost my initial bankroll by not realizing the proper bankroll requirements for playing limit poker.   

A normal string of losses at 5/10 limit poker quickly depleted my $3600 bankroll that I had built over the months preceding that one month run.

I started reading about bankroll requirements and started following the advice I found on them fanatically.  Conservatively would be the way most people would term how I follow those bankroll requirements. 
Mark Blade wrote an excellent book called Professional Poker in which he devotes two chapters to money and bankroll management for professional poker players.  The gist of those chapters that I summarize below is for someone playing out of a given bankroll size for a living.  Thus they do not apply to someone who is not playing for a living and can thus 'reload' his bankroll from other sources of income.   

Additionally, this figure assumes that the professional NEVER takes any money out of his bankroll which is unrealisitc. If you assume that a professional will take out all earnings above 300 big bets every month as profit....then every month he has a 3% chance of going broke and guess what?  Within a relatively short period of time, this player absolutely will go broke! (Almost 1/2 of the time he will be broke within 3 years in this example!) If you are going to take a set amount of expense s every month, I recommend that your initial roll be the amount determined below PLUS your monthly set amount.  Then, I also recommend that you leave any excess earnings above your set expenses in the bankroll at least until you have 12 months worth of expenses in your bankroll so that you can play very comfortably with no or miniscule chance of going broke and so that you can move up in level to make more money. After all aren't you reading this because you want to do this for a living?  I like the idea of a job where I set my hours AND don't have to worry about going broke if I am reasonably conservative.  My conservative bankroll requirements allow me to do that.

For limit poker players he recommends 300 big bets as a standard to ensure that a 1 big bet/hr earning poker player will have a less than 3% chance of going broke with a standard expectation or likelihood of going broke due to 'bad luck'.   Given this standard a $5/10 limit player would need $3000 or 300 times the big bet of $10 to have a 95% certainty of not going broke.   However, a key assumption in this sentence above is '1 big bet/hr earning poker player'.  Many poker players only earn 1/2 bb/hr or 1/4.  If you only earn 1/2 big bet/hr, then you need to double your bankroll to $6,000 to have the same expectation (97%) of not going broke.  If you only earn 1/4 then you need 4x more than the 1bb/hr earning player or $12,000.   

Another thing to consider is your style of play.  A VERY conservative player might be able to get by with $2000 in $5/10 if he is a 1bb/hr winning player.  On the other hand a very aggressive, loose player will have much bigger bankroll swings and need $10,000 in his bankroll to have the same chance (3%) of not going broke.

Next is no limit poker which few people have dealt with in great detail about bankroll management.  Mark Blade recommends 800 big blinds for no limit players and I have to heartily disagree with him here.  800 big blinds in 1/2 NL would be $1600 which is FAR too little to ensure that you have a 3% chance of going broke. 

A VERY good, expert no limit player that is earning 10bb/hr (big blinds in NL not big bets since there are none.) can expect swings easily in excess of 800 big blinds.   I think in terms of max buy ins which is 100bb for most online NL games.  For no limit I would recommend at least 20 buy ins if you are a conservative player making 8-10bb/hr.  This means that $4000 would be a sufficient bankroll to ensure that an expert NL player that is making 8-10bb/hr will have less than a 3% chance of going broke. 

A typical loose aggressive player playing short handed NL can expect much bigger swings and I can attest from personal experience that daily swings as high as $3000 are possible!  (I was also five tabling,  but it gives you an idea why I recommend 20 buy ins.)  I would recommend $8000 for a typical LAG playing 1/2 NL that makes 8-10bb/hr playing 1/2 NL.  Just as in limit, if you are making 1/2 as much in terms of bb/hr playing NL, then you need to double your bankroll requirements.

Regarding MTTs, Blade parrots Sklansky and recommends 55 buy ins to have a 95% chance of never going broke.  That is if you are 150% ROI winning MTT player!  Having well over 1500 MTTs in my personal database, I can attest that it is not unusual for someone who makes 100% ROI and final tables 13% of the time in 200 player size MTTs (that would be me) to have runs of 30 tournaments without a CASH, let alone a final table/real money appearance.  55 buy ins sounds like a reasonable number to me if you have the kind of results that I mention above.  

Of course if your results are half as good then you will need to double your bankroll size to 110 buy ins.    The 50 buy ins that I have seen recommended for SNGs is a fairly reasonable number IF you are able to show at least an in the money (ITM) percentage of 40%.  You have to achieve a 36.5% ITM if you have an average (2nd Place)  finish expectation just to break even in order to cover the rake in a 10 man SNG.

Less than that and I would suggest at least doubling your bankroll requirements or finding a different bankroll building game since you may not be a winning player unless you have a disproportionate number of first place finishes!  A conservative approach to ensure a very low chance of going broke (3% or less) would suggest not moving up in SNG levels until achieving 100 buy ins at the level you moving to!

Best of luck out there on the tables!