By transforming a central limit theorem for dependent variables, we find conditions for a sequence of processes with paths in $D\lbrack 0, \infty)$ to converge weakly to a diffusion process. Of the most important conditions, the first is related to (but weaker than) tightness, and in the next two we require that the first two conditional moments, given the past, of truncated increments in small time intervals, should stay close to the appropriate infinitesimal coefficients of the limiting diffusion times the length of the time interval. The limiting diffusions can have inaccessible or exit boundaries. We prove that our conditions are necessary and sufficient in order that: (1) the sequence of processes converges weakly in $D\lbrack 0, \infty)$; (2) any finite number of conditional expectations given the past of bounded, continuous functionals of the processes converge jointly in distribution to the "correct" value.