We study exact dimensionality and dimension formulas for invariant measures in finite or countable conformal iterated function systems with overlaps, and also in random systems. First we recall the exact dimensionality result of (Feng and Hu in Commun Pure Appl Math 62:1435–1500 [17]) for invariant measures in finite conformal iterated function systems (IFS) with overlaps. Then we give another exact dimensionality result, in particular another proof for self-conformal measures, and a dimension formula which uses overlap numbers. Next we present the case of random countable conformal IFS with overlaps, and we show that the projection of any ergodic measure with finite entropy is exact dimensional, and obtain a dimension formula. In particular this proves the exact dimensionality in deterministic countable IFS with overlaps. Then, we give the notion of Smale endomorphisms over shifts with countable alphabets. We prove that the projections of conditional measures of equilibrium states of summable Hölder continuous potentials are exact dimensional on fibers. Moreover the global push-forward measures are shown to be exact dimensional, and a general dimension formula is obtained. This applies to natural extensions of the continued fractions transformation, and of \(\beta \) -maps for \(\beta >1\) . It allows to extend the Doeblin-Lenstra Conjecture in Diophantine approximation. We then study a class of transformations with singularities, related to multidimensional continued fractions.