\section{Concluding Remarks}
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Heavy quark physics, in particular the physics of the bottom quark, 
will provide a deeper insight into the question of quark mixing and 
the origin of $CP$ violation. The possibility of testing this sector 
of the standard model with unprecendented precision has motivated 
a very strong effort of a large part of the particle physics community.

In particular from the experimental side impressive progress has been 
made, mainly due to technological progress, among which the development 
of silicon vertex detectors was the most prominent. It is mainly this 
piece of equipment which allowed the LEP experiments to compete with 
the present dedicated $B$ physics experiments such as CLEO and ARGUS. 
This fact is very astonishing in retrospect, since by the time LEP
started no one would have guessed that e.g. the $B$ hadron lifetime 
measurements would be dominated by the LEP experiments. 

The future efforts from the experimental side spans practically all 
possibilities of how to study $b$ hadrons. On one hand there are 
three dedicated $e^+ e^-$ machines running at energies about the 
$\Upsilon(4s)$: The CLEO experiment located at the symmetric machine 
CESR at Cornell, and two asymmetric $B$ factories, one at SLAC (the 
BaBar experiment) and one at KEK (the BELLE experiment). The latter 
two are designed in an asymmetric way in order to make a time dependent 
measurement of the $CP$ asymmetries in the $B_d$ system possible. 

On the other hand there are experiments based on hadronic collisions. 
There is one fixed target experiment planned at DESY in Hamburg 
(the HERA-B experiment) which has as its main target the $CP$ asymmetries
in the decay $B_d \to J/\Psi K_s$ and the CDF experiment at FERMILAB 
is now upgraded to allow for measurements of bottom hadron decays. 
In the longer run  a dedicated $b$ physics experiment is planned 
at the LHC, the LHC-B experiment, which will due to the very large 
statistics allow additional measurements. 

Parallel to these experimental efforts there has been a breakthrough 
in the theoretical description. Many of the calculations of weak processes
of heavy quarks are meanwhile based on the $1/m_Q$ expansion of 
QCD and the model uncertainties have been substantially reduced. 
The purpose of the present review was to summarize the theoretical 
development based on the heavy mass expansion using the methods of 
effective field theory. 

Ultimately all the unknown nonperturbative quantities will be evaluated  
on the lattice, still the heavy mass expansion has turned out to be a 
useful tool to obtain a better understanding of the lattice data. At 
present the computer power is still too limited to calculate e.g. the 
form factors of a weak transition over the full kinematic range, but 
first steps in this direction exist.   

  
\subsection{Conclusions and Open Questions: Exclusive Decays}
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As far as exclusive decays are concerned the theoretical description 
seems to be in quite good shape. The master example is the determination 
of $V_{cb}$ from the $q^2$ spectrum of the leptons in 
$B \to D^* \ell \bar{\nu}_\ell$, where heavy quark symmetries allow 
access to the absolute value of the relevant form factor at maximum $q^2$. 
Uncertainties enter here through new unknown matrix elements at the level 
of $1/m_c^2$, which have been estimated to be about five percent and hence 
quite small. On the other hand, with present theoretical technology 
this seems to be a serious limitation which will be hard to overcome. 
The expectation from the future $B$ factories 
on the uncertainties $V_{cb}$ is that by the end of the millenium $V_{cb}$ 
will be known with an uncertainty of five percent. This has to be compared 
to our present knowledge of the Cabibbo angle, i.e. $V_{us}$ which is 
known at a precision of about one percent.   

Exclusive semileptonic heavy to light transitions are not as strongly 
retricted by heavy quark symmetries. Still relative normalizations are
possble which are, however, not protected against $1/m_Q$ corrections.
In spite of this, exclusive heavy to light decays will play some role 
in the determination of $V_{ub}$, but this requires  measurement of the 
lepton spectrum in $b \to u$ decays. The determination of $V_{ub}$ --
at least judging from present theoretical technology -- has to rely 
on mix of different methods, including also inclusive decays. A realistic 
estimate on the theoretical uncertainty on $V_{ub}$, which can be achieved 
with present methods, is ten percent.
 
Exclusive nonleptonic decays remain the most difficult kind of processes
despite of the heavy quark limit. In exclusive semileptonic decays heavy 
quark symmetries allow us in some cases to ged rid of the details of QCD
dynamics and hence get access to weak interaction quantities without 
``solving'' QCD. However, in nonleptonic decays QCD plays a crucial role
and there is no way to employ heavy quark symmetries to ``hide'' the strong
interaction problems. 
 

\subsection{Conclusions and Open Questions: Inclusive Decays}
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In order to set up a $1/m_Q$ expansion for inclusive decays one makes 
use of the fact that the mass of the heavy quark set a large scale and 
performs an operator product expansion. For semileptonic decays one need 
to assume global duality (i.e. that partonic and hadronic quantities 
coincide in the average) which is generally believed to be safe 
assumption. For nonleptonic processes one need the stronger assumption 
of local duality (i.e. that hadronic and partonic quantities concide
at some kinematic point) and a part of the community believes that this 
may be the reason for the problems in describing the lifetimes, although 
local duality is assumed at a point far away from the resonance region.

One may also calculate the lepton energy spectrum in inclusive semileptonic 
decays using the operator product expansion. In this case the expansion 
parameter is the energy release $m_Q - 2 E_\ell$ which is in almost all phase 
space of the order of the heavy quark mass. However, close to the maximal
energy this expansion breaks down, yielding unphysical spikes in the spectrum.
However, in the endpoint region, which is assumed to be wide enough to still 
employ duality, one may perform a resummation of the most singular terms, 
yielding a distribution function for the light cone residual momentum of the 
heavy quark inside the heavy meson, in close analogy to what happens in 
deep inelastic scattering. Even closer to the endpoint one enters the 
resonance region, where the spectrum is dominated by the lowest resonances
which can appear as final states in the inclusive process.  

Both CKM parameters $V_{ub}$ and $V_{cb}$ may be determined from inclusive 
decays. As far as $V_{cb}$ is concerned, the inclusive method can compete 
with the exclusive one, since the theoretical uncertainties are comparable. 
It is already a nontrivial check that the values of $V_{cb}$ determined from
inclusive and exclusive processes indeed coincide very nicely. $V_{ub}$ has 
been determined from the endpoint region of the inclusive decay 
$B \to X_u \ell \bar{\nu}_\ell$ looking at events beyond the endpoint of 
charmed final states. With improved technology it may be possible to 
determine the hadronic invariant mass of the final state and hence allow us 
to obtain a better determination of $V_{ub}$ at future $B$ factories.  

\subsection{Outlook}
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In this review considered the implications of heavy quark symmetry 
and the $1/m_Q$ expansion of QCD without using additional input
It is obvious that one may add on top  of heavy quark 
symmetries other methods which have been developed in QCD. One additional 
input which has been discussed is to implement also the chiral symmetry 
of the light degrees of freedom \cite{HQchi} which has provided some 
insight e.g. into the form factors of heavy light transitions. 

Another input to supplement the nonperturbative information could be QCD
sum rules or even models. In particular the latter may be checked, since 
a simple requirement for models is compatibility with heavy quark 
symmetries. The ultimate way to determine nonperturbative parameters 
are lattic calculations which eventually will not use the heavy mass 
expansion but rather deal with full QCD. 

Recently heavy quark symmetries, lattice data and constraints from unitarity 
and analyticity have been combined to contrain form factors \cite{anaform}. 
In particular for heavy to light transitions this input basically fixes the 
possible shape of the form factors and will eventually yield a reliable way
to determine $V_{ub}$ from exclusive decays.

\section*{Acknowledgements} 
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I thank C. Balzereit for a careful reading of the manuscript. 




