Next generation of wireless communication mostly relies on multiple-input multipleoutput (MIMO) configuration and full-duplex relaying to improve data-rates, spectrale efficiency, spatial-multiplexing, quality-of-service and energy-efficiency etc. However, multiple radio frequency (RF) transceivers in MIMO system and multi-hops in relay networks, accumulate transceiver impairments, rendering an unacceptable system performance. Majority of the technical contributions either assume ideal hardware or inappropriately model hardware impairments which often induce misleading results especially for high data-rate communication systems. We propose statistical mathematical modeling of various hardware impairment (HWI) to characterize their deteriorating effects on the information signal. In addition, we model the aggregate HWI as improper Gaussian signaling (IGS), to fully characterize their asymmetric properties and the self-interfering signal attribute under I/Q imbalance. The proposed model encourages to adopt asymmetric transmission scheme, as opposed to traditional symmetric signaling. First, we present statistical baseband equivalent mathematical models for general MIMO system and two special scenarios of receive and transmit diversity systems under HWI. Then, we express their achievable rate under PGS and IGS transmit schemes. Moreover, we tune the IGS statistical characteristics to maximize the achievable rate. We also present optimal beam-forming/pre-coding and receive combiner vector for multiple-input single-output (MISO) and single-input multiple output (SIMO) systems, which lead to SDNR maximization. Moreover, we propose an adaptive scheme to switch between maximal IGS (MIGS) and PGS transmission based on the described conditions to reduce computational overhead. Subsequently, two case studies are presented. 1) Outage analysis has been carried out for SIMO, under transceiver distortion noise, for two diversity combining schemes 2) The benefits of employing IGS is investigated in full duplex relaying (FDR) suffering from two types of interference, the residual self-interference (RSI) and I/Q distortions. We further optimize the pseudo-variance to compensate the interference impact and improve end-to-end achievable rate. Finally, we validate the analytic expressions through simulation results, to quantify the performance degradation in the absence of ideal transceivers and the gain reaped from adopting IGS scheme compared with PGS scheme.