We have synthesized polycrystalline and single-crystal samples of PbCu3TeO7 and studied its properties via magnetic susceptibility, χ(T), and heat-capacity, Cp(T), measurements and also electronic structure calculations. Whereas the crystal structure is suggestive of the presence of a quasi-2D network of Cu(2+) (S = 1/2) buckled staircase kagome layers, the χ(T) data show magnetic anisotropy and three magnetic anomalies at temperatures TN1 ∼ 36 K, TN2 ∼ 25 K, and TN3 ∼ 17 K. The χ(T) data follow the Curie-Weiss law above 200 K and a Curie-Weiss temperature θCW ∼- 150 K is obtained. The data deviate from the simple Curie-Weiss law below 200 K, which is well above TN1, suggesting the presence of competing magnetic interactions. The magnetic anomaly at TN3 appears to be of first order from magnetization measurements, although our Cp(T) results do not display any anomaly at TN3. The hopping integrals obtained from our electronic structure calculations suggest the presence of significant intra-kagome (next-nearest neighbor and diagonal) and inter-kagome couplings. These couplings take the PbCu3TeO7 system away from a disordered ground state and lead to long-range order, in contrast to what might be expected for an ideal (isotropic) 2D kagome system.