Over the past few years, learning-based video compression has become an active research area. However, most works focus on P-frame coding. Learned B-frame coding is under-explored and more challenging. This work introduces a novel B-frame coding framework, termed B-CANF, that exploits conditional augmented normalizing flows for B-frame coding. B-CANF additionally features two novel elements: frame-type adaptive coding and B*-frames. Our frame-type adaptive coding learns better bit allocation for hierarchical B-frame coding by dynamically adapting the feature distributions according to the B-frame type. Our B*-frames allow greater flexibility in specifying the group-of-pictures (GOP) structure by reusing the B-frame codec to mimic P-frame coding, without the need for an additional, separate P-frame codec. On commonly used datasets, B-CANF achieves the state-of-the-art compression performance as compared to the other learned B-frame codecs and shows comparable BD-rate results to HM-16.23 under the random access configuration in terms of PSNR. When evaluated on different GOP structures, our B*-frames achieve similar performance to the additional use of a separate P-frame codec.