We study contractions of affine Kac-Moody algebras (KMAs) with respect to their twisted and untwisted Kac-Moody subalgebras and also contract twisted KMA with respect to their affine subalgebras. Our work generalizes the Inönü-Wigner contraction of KMA by Majumdar [J. Math. Phys. 34, 2059 (1993)] in several directions. We compare Inönü-Wigner contractions with respect to twisted subalgebras to another kind of contractions, which we call modulo contractions. We also introduce a novel type of contractions, based on Z-gradings of KMA, which we call level contractions. Along the way we use an algorithm for constructing "Kac-Moody-like algebras" g for any (and not just semisimple) finite-dimensional Lie algebra g.