In this thesis, we are concerned with the problem of counting algebraic points of bounded height and degree on graphs of certain transcendental holomorphic and meromorphic functions. Adopting a Nevanlinna theoretic approach for the latter, we attain bounds of the form \(C(d)(\log H)b\) for the number of algebraic points of height at most H and degree at most d on the restrictions to compact subsets of domains of holomorphy of meromorphic functions with certain growth/decay conditions. In the second half of the thesis, we turn our attention to counting points on graphs of certain analytic functions with growth behaviour stricter than finite order and positive lower order. For these functions, we are able to relax the need to restrict them to compact subsets of \(C\), and indeed, to count points either on the whole graph or nearly all of it. For these functions we also attain a bound of the form \(C(d)(\log H)b\). We end this work with several pointers towards possible extensions of our results. The results in this thesis can be seen as extensions of the work of Boxall and Jones on algebraic values of certain analytic functions.