Biobanks store and catalog human biological material that is increasingly being digitized using next-generation sequencing (NGS). There is, however, a computational bottleneck, as existing software systems are not scalable and secure enough to store and process the incoming wave of genomic data from NGS machines. In the BiobankCloud project, we are building a Hadoop-based platform for the secure storage, sharing, and parallel processing of genomic data. We extended Hadoop to include support for multi-tenant studies, reduced storage requirements with erasure coding, and added support for extensible and consistent metadata. On top of Hadoop, we built a scalable scientific workflow engine featuring a proper workflow definition language focusing on simple integration and chaining of existing tools, adaptive scheduling on Apache Yarn, and support for iterative dataflows. Our platform also supports the secure sharing of data across different, distributed Hadoop clusters. The software is easily installed and comes with a user-friendly web interface for running, managing, and accessing data sets behind a secure 2-factor authentication. Initial tests have shown that the engine scales well to dozens of nodes. The entire system is open-source and includes pre-defined workflows for popular tasks in biomedical data analysis, such as variant identification, differential transcriptome analysis using RNA-Seq, and analysis of miRNA-Seq and ChIP-Seq data.