  The definition for the $p$-adic Hurwitz-type Euler zeta functions has been
given by using the fermionic $p$-adic integral on $\mathbb Z_p$. By computing
the values of this kind of $p$-adic zeta function at negative integers, we show
that it interpolates the Euler polynomials $p$-adically. Many properties are
provided for the $p$-adic Hurwitz-type Euler zeta functions, including the
convergent Laurent series expansion, the distribution formula, the functional
equation, the reflection formula, the derivative formula, the $p$-adic Raabe
formula and so on. The definition for the $p$-adic Euler $L$-functions has also
been given by using the $p$-adic Hurwitz-type Euler zeta functions.
