Usually, when we are dealing with approximations of an irrational α by rational fractions p/q, we want to solve the system of inequalities |α q - p| < ε, 1≤ q ≤ Q in integers p,q. This formulation leads to ordinary continued fractions. Similar formulations corresponding to the L₂ and L₁-norms go back to Hermite and Minkowski. They are related to other types of continued fractions. We will discuss these algorithms and explain how these constructions are related to the relatively new concept of Dirichlet improvability. In particular, we will answer some questions by D. Kleinbock. This is a joint work with Nikita Shulga.