Abstract: This dissertation concerns existence of a steadily translating finger solution in a Hele-Shaw cell for small but non-zero surface tension ($\epsilon^2 $).Though there are numerous numerical and formal asymptotic results for this problem, we know of no mathematically rigorous results to date. By showing equivalence of the problem to a certain nonlinear integro-differential equation in a complex plane strip, we show that analytic symmetric finger solutions exist for sufficiently small surface tension for a discrete set of values of $(\lambda -\frac{1}{2})/\epsilon^{\frac{4}{3}}$, where $\lambda$ is the relative finger width. The methodology consists of proving existence and uniqueness of analytic solutions for a weaker half strip problem for any $\lambda$ where the symmetric condition is relaxed. This is followed by analysis near appropriate complex turning points that determine exponentially small terms in $\epsilon$,as $\epsilon \to 0^+$. We prove that for $\lambda-\frac{1}{2}$ small the symmetry condition is satisfied if only if $(\lambda -\frac{1}{2})/\epsilon^{\frac{4}{3}} $ attains a discrete set of values. When the symmetry condition is satisfied,the weaker half strip problem is shown to be equivalent to the original steady finger problem.