Blue compact galaxies (BCGs) are gas-rich star-forming low redshift galaxies with low metallicities. In some cases the relative strength of the starburst can be so high that it completely dominates the light output of the galaxy, an obstacle which has been countered by deeper optical imaging data and observations in the near infra-red (NIR) regime. This has revealed an older population referred to as the "host". In an effort to study the hosts of BCGs we have analyzed new and extremely deep UBVRIHKs imaging data for 46 high and low luminosity BCGs. For several BCGs the data reveal previously undetected extended low surface brightness components beyond the μB~26 mag arcsec-2 isophote. These are predominantly the luminous BCGs in the sample, and they show tails, plumes, optical bridges between companion galaxies, and other signs of merging or strong tidal interactions. The low luminosity BCGs, on the other hand, are well represented by an exponential disk profile down to the reliability limit of the data at a surface brightness level of μB~28 mag arcsec-2. The burst and host populations are examined separately. The integrated colors of both are compared to predictions from spectral evolutionary models, giving an indication of their respective ages and metallicities. Our analysis suggests that for the luminous BCGs a strong contribution by nebular emission is present almost down to the Holmberg radius, invalidating the host structural parameters obtained from brighter isophotes. Possible evolutionary links to quiescent galaxies like dEs, dIs, and LSBGs are explored by examining the structural parameters derived from two radial ranges typically assumed to be dominated by the underlying host galaxy. In this parameter space the luminous BCGs in our sample deviate from their low luminosity counterparts and from BCG data in the literature. They are instead consistent with the structural properties of giant low surface brightness galaxies with central surface brightnesses μB≥23 mag arcsec-2. We further examine the asymmetry and concentration parameters for the sample and study the correlation between the minimum asymmetry distribution in the optical and NIR vs morphological class, concentration and integrated colors to identify mergers/tidally interacting galaxies. A shift in the asymmetry distribution occurs for low luminosity BCGs from the optical to the NIR. In contrast, we find that the flocculent asymmetry component (due to star formation) completely dominates the composite asymmetry of high luminosity BCGs. We introduce an alternative asymmetry measure which successfully traces the dynamical asymmetry component (due to merging/tidal interaction) of the host. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Manuscript. Paper 5: Manuscript.</p>