Keeping up in technology
From EU3 Wiki
There are 5 areas of research in this game. Goverment, Production, Trade, Naval and Land.
Consult the Tech Tree to see what is the advantage of going to the next level in each area. Also, keep in mind that being too far ahead is penalised heavily so you should avoid being too far ahead in one area and instead try to balance your research spending your money on a few areas at once, this way you won't go too far in one area and won't have to pay the penalty.
Instead of minting ducats you should reinvest all your monthly incomes into your research and finance your wars, colonization and diplomatic efforts with the money you get annually.
It is wise to decide in which areas you want to be strong and which you can abandon. Inland countries can ignore Naval tech, those that wage many wars, have low stability and reputation all the time, can safely ignore Trade, peaceful guys can abandon Land and so on, use your common sense here.
Thanks to that you can save money in one area and spend them on another, more important one.
The number of provinces a country controls influences costs of each level of each technology. Countries that control many provinces will have to spend many times more money than those which control only a few provinces.
It doesn't pay off though to stay small. It pays off to expand but you have to do it in a smart way. Countries that lag behind in research are usually those which have many very poor provinces. Those that expand into rich areas, like Portugal and Castille colonizing many provinces producing sugar, coffee, tobacco and cotton, are always far ahead in the means of research. So expand carefully and don't be too greedy to have more and more land. What you need are provinces that can afford themselves and this usually depends on their production revenue. So avoid anything that produces slaves, wheat, fish, naval supplies and such and go for gold and other great goodies instead.
Trading can double or even triple your research rate. So don't ignore this aspect of the game, although it may seem very frustrating. Send your merchants to the richest centers of trade and make sure they stay there for long. Just make sure you have Shrewd Commerce Practice and National Trade Policy national ideas. Also, do not ignore your Trade research, with each level of it your merchants live longer and earn more money.
Your neighbours help you in your research efforts. You get additional investment every time they outpace you.
There is an advisor specific for each research area that can improve your investment significantly. For most countries it is crucial to hire such advisors before AI takes them over.
Your monarch can boost your research in certain areas depending on his skills.
Finally, you can build a building (or take over a province which has it built) specific for each area that boosts research in it. Universities add +5 to your Goverment research, Weapon Manufactories boost your Land research and so on. All such buildings are listed on the Province improvements page.
It is worth noting that your research depends heavily on the tech group you belong to. We have a separate page describing all the Tech groups.
If you belong to a group that is very backwards in the means of research you may consider Westernisation that moves your country to the best (Latin) group if it's possible. If not, you just have to get used to the backwardness of your country.

