Two-dimensional (2 D) electron systems in compound semiconductors are markedly important not only for studies of low-dimensional electron physics but also for application in recent information-oriented society. To investigate behavior of such 2 D electrons in nanometer-scale regions, the local density of states (LDOS) was characterized at the clean surfaces in ultra-high vacuum using a low-temperature scanning tunneling microscope (LT-STM). This LT-STM was equipped with a chamber for molecular beam epitaxy (MBE) of III-V compound semiconductors. At the InAs(111)A surfaces, LDOS standing waves were clearly imaged around the defects. The wave features were confirmed to originate from the Friedel oscillation of naturally formed 2 D states in the surface electron accumulation layer. Also, the electronic states of highly Si-doped InGaAs surface quantum wells (QWs), grown on the GaAs(111)A substrates, were characterized. The LDOS spectra revealed the formation of artificial 2 D subbands quantized in the QWs. The effects related to phase coherence in the QWs are discussed. It is shown that nanoprobing the clean surfaces is very useful for microscopic understanding of quantum mechanical phenomena aiming at advanced electronic devices for future communication technologies.