The κ-μ / inverse gamma and η-μ / inverse gamma composite fading models are presented and extensively investigated in this paper. We derive closed-form expressions for the fundamental statistics of the κ-μ / inverse gamma composite fading model, such as the probability density function (PDF), cumulative distribution function (CDF), moment generating function (MGF), higher order moments and amount of fading (AF). Closed-form expressions for the PDF, higher order moments and AF are also obtained for the η-μ / inverse gamma composite fading model while infinite series expressions are obtained for the corresponding CDF and MGF. The suitability of the new models for characterizing composite fading channels is demonstrated through a series of extensive field measurements for wearable, cellular and vehicular communications. For all of the measurements, two propagation geometry problems with special relevance to the two new composite fading models, namely the line-of-sight (LOS) and non-LOS (NLOS) channel conditions, are considered. It is found that both the κ-μ / inverse gamma and η-μ / inverse gamma composite fading models provide an excellent fit to fading conditions encountered in the field. The goodness-of-fit of these two composite fading models is also evaluated and compared using the resistor-average distance. As a result, it is shown that the κ-μ / inverse gamma composite fading model provides a better fit compared to the η-μ / inverse gamma composite fading model when strong dominant signal components exist. On the contrary, the η-μ / inverse gamma composite fading model outperforms the κ-μ / inverse gamma composite fading model when there is no strong dominant signal component and/or the parameter η is not equal to unity, indicating that the scattered wave power of the in-phase and quadrature components of each cluster of multipath are not identical.