We review the euclidean path-integral formalism in connection with the
one-dimensional non-relativistic particle. The configurations which
allow construction of a semiclassical approximation classify themselves
into either topological (instantons) and non-topological (bounces)
solutions. The quantum amplitudes consist of an exponential associated
with the classical contribution as well as the energy eigenvalues of the
quadratic operators at issue can be written in closed form due to the
shape-invariance property. Accordingly, we resort to the zeta-function
method to compute the functional determinants in a systematic way. The
effect of the multi-instantons configurations is also carefully considered.
To illustrate the instanton calculus in a relevant model, we go to the
double-wall potential. The second popular case is the periodic-potential
where the initial levels split into bands. The quantum decay rate of the
metastable states in a cubic model is evaluated by means of the bounce-like
solution.