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Notes: Summary. Red giants are sometimes surrounded by envelopes, the result of the ejection of stellar matter at a large rate ( $\dot M〉 10^{-7}M_\odot$ /yr) and at a low velocity (10 km/s). In this review the envelopes are discussed and the relation between stars and envelope: what stars combine with what envelopes? The envelope emits radiation by various processes and has been detected at all wavelengths between the visual and the microwave range. I review the observations of continuum radiation emitted by dust particles and of rotational transitions of molecules, where these molecules have been excited by thermal or by non–thermal (“maser”) processes. I discuss mainly the oxygen–rich stars, those of spectral type M, and only briefly the closely related carbon–rich stars. By and large the density in the envelope is well described by spherically symmetric outflow at a constant velocity; on the time scale needed to flow from stellar surface to the outermost layers, i.e. $10^5$ yr, the loss of mass is sometimes interrupted suddenly after which the envelope becomes “detached” from the star. The temperature decreases when moving outward; heat input is by friction between dust particles and gas and cooling occurs by line radiation by various molecules, especially by H $_2$ O. The molecular composition is determined by formation in an equilibrium process deep in the atmosphere and by destruction in the outer parts of the outflow by interstellar UV radiation (H $_2$ , CO, H $_2$ O) or by depletion due to condensation on dust grains (SiO); dust particles of silicate material solidify where the radiation temperature is decreased to about 1000 K, and this is at a few stellar radii. The various continuum spectra produced by the dust particles in different stars are well modelled by a simple model of the density and dust temperature distribution plus the assumption that the particles consist of “dirty silicate”, i.e. silicate with Fe and Al ions added. A large range of optical depths, $\tau_{9.7\mu{\rm m}}$ , is observed: from 0.01 to 10. In envelopes with large optical depth the star itself can no longer be detected directly. Model calculations also show that the momentum in the outflow, i.e. $\dot M.v_{\rm out}$ is provided by radiation pressure on the dust particles followed by the complete transfer of this momentum to the gas. The mass–loss rate itself, $\dot M$ , is not determined by radiation pressure but by dynamic processes in the region below the dust condensation layer. When $\tau_{9.7\mu{\rm m}}$ is sufficiently large its measurement, that of the stellar luminosity, $L_*$ and that of the outflow velocity, $v_{\rm out}$ , permit the determination of $\dot M$ , i.e. the total outflow rate, without making assumptions about the abundance of the dust particles or of the molecular gases. Detached envelopes have been seen in a few cases. Thermal molecular radiation is faint compared to the maser emission but has been measured in distant stars, e.g. in stars near the galactic center. Different molecules outline different “spheres” around the star. The largest sphere (a radius of 0.1 pc) is outlined by an emission line belonging to the CO( $v=0, J=1\to 0$ ) transition. Higher rotational transitions of CO give smaller diameters. A comparison of CO ( $J=2\to 1$ ) and ( $J=1\to 0$ ) fluxes in stars with very thick envelopes leads to the conclusion that an abrupt decrease in the mass–loss rate occurred some ten thousand years ago. Three molecules produce each several maser lines: SiO, H $_2$ O and OH. Several new H $_2$ O lines have recently been discovered; their exploration has hardly been started. The high intensity of the maser lines makes interferometry possible and hence detailed mapping. The SiO lines are formed deep in the envelope, below the dust condensation layer. OH maser lines are produced farthest out, H $_2$ O lines in between. The excitation mechanisms for most maser lines is understood globally, but detailed models are lacking, largely because the problem is non–linear and the solution of the radiative transfer equation requires a highly anisotropic geometry. The geometrical and kinematical properties of the 1612 MHz OH maser, which in many objects is very strong, are explained by a thin shell of OH; because the angular diameter of the shell can be measured directly and the linear diameter can be determined from the difference in the time of maximum flux of blue and red maser peaks, the distance of the shell and of the star can be measured. The presence or absence of individual maser lines appears to depend on the value of $\tau_{9.7\mu{\rm m}}$ and is well described by a sequence called “Lewis' chronology”. The central star is a long–period variable with a period of 300 days or longer and with a large luminosity amplitude ( $\Delta m_{\rm bol}〉 0.7^m$ ). Evidence is given that each star has the maximum luminosity it will reach during its evolution and that it is a thermally–pulsing Asymptotic–Giant–Branch star (TP–AGB) with a main–sequence mass between 1 and 6 $M_\odot$ . Stars of the same main–sequence mass, $M_{\rm ms}$ , have different mass–loss rates, in some cases by a factor of 10. The mass–loss rate probably increases with time, and the highest mass–loss rates are reached toward the end of the evolution. Stars with higher $M_{\rm ms}$ ultimately reach higher mass–loss rates. The calibration of the main–sequence mass is reviewed. Most Mira variables with mass loss have a mass between 1.0 and 1.2 $M_\odot$ . OH/IR stars with periods over 1000 days have no counterparts among the carbon stars and thus have $M_{\rm ms}〉 4.5M_\odot$ . Stars as discussed in this review have been found only in the thin galactic disk and in the bulge. Finally I review several recently proposed scenarios for TP–AGB evolution in which mass loss is taken into account. These scenarios represent the observations quite well; their major short–coming is the lack of an explanation why the central stars are always large–amplitude, long–period variables and why such stars are the ones with high mass–loss rates.