The study is about metric and arithmetic properties of algorithms for continued fractions, which are used for approximating real numbers by rational numbers. The first part is on approximation coefficients of regular continued fractions. These coefficients indicate the quality of an approximation. The study is focused on examining properties of these coefficients. Next, I will study the recently introduced N-expansions, which in some ways have more attractive properties than regular expansions but may lack some other. Finally, I will focus on a/b-continued fractions, based on a recently developed algorithm for continued fractions.