Natural (red blood cells) and artificial biconcave-discoid-shaped capsules have immense biological (a cellular component of blood) and technological (as drug carrier) relevance, respectively. Their low reduced volume allows significant shape changes under external fields such as extensional flows (encountered at junctions and size-varying capillaries in biological flows) and electric fields (in applications such as electroporation and dielectrophoresis). This work demonstrates biconcave-discoid to capped-cylindrical and prolate-spheroid shape transitions of a capsule in uniaxial extensional flow as well as in DC and AC electric fields. The shape changes of a stress-free biconcave-discoid capsule in external fields are important in determining the momentum and mass transfer between the capsule and the medium fluid as well as dielectrophoresis and electroporation phenomena of a capsule in an electric field. The biconcave-discoid to capped-cylindrical/prolate-spheroid shape transition is demonstrated for both a capsule (with parameters relevant to drug delivery) as well as for a red blood cell (physiological conditions). However, significant differences are observed in this shape transition depending upon the applied external fields. In an extensional flow, the pressure-driven transition shows the equator being squeezed in and the poles being pulled out to deform into a capped cylinder at low capillary number and a prolate spheroid at high capillary number. On the other hand, in the transition driven by electric fields, the shoulders of the capsule seem to play a significant role in the dynamics. The shape transition in the electric fields depends upon the relative magnitude of the electric and the hydrodynamic response times, particularly relevant for the dynamics of red blood cells in physiological conditions. A new method of analysing the shape transition of red blood cells in AC electric fields is suggested, where a large separation of time scales is observed between the hydrodynamic and electric responses.