  We present four generalized small cancellation conditions for finite
presentations and solve the word- and conjugacy problem in each case. Our
conditions $W$ and $W^*$ contain the non-metric small cancellation cases C(6),
C(4)T(4), C(3)T(6) (see [LS]) but are considerably more general. $W$ also
contains as a special case the small cancellation condition $W(6)$ of Juhasz
[J2]. If a finite presentation satisfies $W$ or $W^*$ then it has a quadratic
isoperimetric inequality and therefore solvable word problem. For the class $W$
this was first observed by Gersten in [G7] which also contains an idea of the
proof. Our main result here is the proof of the conjugacy problem for the
classes $W$ and $W^*$ which uses the geometry of non-positively curved
piecewise Euclidean complexes developed by Bridson in [Bri]. The conditions $V$
and $V^*$ generalize the small cancellation conditions C(7), C(5)T(4),
C(4)T(5), C(3)T(7). If a finite presentation satisfies the condition $V$ or
$V^*$, then it has a linear isoperimetric inequality and hence the group is
hyperbolic.
