In [Fuzzy Sets Syst. 94, No. 3, 405-410 (1998; Zbl 0936.54013)] the author introduced a notion of compactness for fuzzy subsets in fuzzy convergence spaces in the sense of [E. Lowen, R. Lowen and P. Wuyts, ibid. 40, No. 2, 347-373 (1991; Zbl 0728.54001)]. The aim of this paper is to investigate Arzela-Ascoli type compactness criteria for fuzzy function spaces in the category FCS of fuzzy convergence spaces. (This category is known to be Cartesian closed.) To obtain such criteria the author first defines a fuzzy convergence structure p-lim of pointwise convergence in fuzzy function spaces for which compactness criteria can be easily established (via the Tikhonov product theorem proved earlier by the author), and then he considers fuzzy subsets of fuzzy function spaces on which structure p-lim coincides with the “standard” convergence structure c-lim on fuzzy function spaces; such fuzzy subsets are called evenly continuous in the paper. (Generally p-lim may be coarser than c-lim).