Topological photonics has attracted widespread research attention in the past decade due to its fundamental interest and unique manner in controlling light propagation for advanced applications. Paradigmatic approaches have been proposed to achieve topological phases including topological insulators in a variety of photonic systems. In particular, photonic lattices composed of evanescently coupled waveguide arrays have been employed conveniently to explore and investigate topological physics. In this article, we review our recent work on the demonstration of topological phenomena in reconfigurable photonic lattices established by site-to-site cw-laser-writing or multiple-beam optical induction in photorefractive nonlinear crystals. We focus on the study of topological states realized in the celebrated one-dimensional Su-Schrieffer-Heeger lattices, including nonlinear topological edge states and gap solitons, nonlinearity-induced coupling to topological edge states, and nonlinear control of non-Hermitian topological states. In the two-dimensional case, we discuss two typical examples: universal mapping of momentum-space topological singularities through Dirac-like photonic lattices and realization of real-space nontrivial loop states in flatband photonic lattices. Our work illustrates how photorefractive materials can be employed conveniently to build up various synthetic photonic microstructures for topological studies, which may prove relevant and inspiring for the exploration of fundamental phenomena in topological systems beyond photonics.