  We review the euclidean path-integral formalism in connection with the
one-dimensional non-relativistic particle. The configurations which allow to
construct a semiclassical approximation classify themselves into either
topological (instantons) and non-topological (bounces) solutions. The quantum
amplitudes consist on an exponential associated with the classical contribution
multiplied by the fluctuation factor which is given by a functional
determinant. The eigenfunctions 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-well 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.
