Electrospinning was used in innovative electrospinning rigs to obtain tubular and flat fibrous structures with controlled fiber orientation with the aim to be used as scaffolds for biomedical applications, more specifically in the tissue engineering of vascular and orthopedic grafts. Gelatine and hydroxyapatite (HA)–gelatine solutions of various compositions were tried and electrospinning of continuous fibers was maintained for gelatine and up to 0.30 g/g HA–gelatine solutions in 2,2,2-trifluoroethanol (TFE). Small diameter tubular scaffolds were electrospun with axial fiber orientation and flat scaffolds were cut from fiber mats electrospun around a wired drum substrate. The fibrous mats were crosslinked using a glutaraldehyde solution and subjected to image analysis of SEM micrographs, water swelling tests, and mechanical testing. Fiber diameter in the electrospun scaffolds could be varied depending on the feed solution concentration and composition whereas fiber orientation was affected by the processing conditions. After crosslinking, the 0.30 g/g HA–gelatine scaffolds absorbed the minimum amount of water after 48 h soaking and they had the highest Young’s modulus, 60 MPa, and highest strength, 3.9 MPa.