In this thesis, we study two problems in random matrices and orthogonal polynomials. The first problem is related to the probability that all the eigenvalues of an n × n Hermitian random matrix from the generalised Gaussian unitary ensemble (gGUE) are positive. To compute this probability, we study the recurrence coefficients of a family of semi-classical Laguerre polynomials { Pₙ(z)} orthogonal with respect to the weight of the form xᵃe⁻ⁿ⁽ˣ⁺ˢ⁽ˣ²⁻ˣ⁾⁾, x ∈ [0, ∞), where α >-1, s ∈ [0, 1] and N > 0. Based on the ladder operators, we show the recurrence coefficients satisfy a second order ordinary differential equation when viewed as functions of the parameter s. Then we work out the large-degree asymptotics of the recurrence coefficients by make use of Dyson's Coulomb fluid approximation theory. We also discuss the associated Hankel determinant Dₙ(s). As the log-derivative of Dₙ(s) can be expressed in terms of the recurrence coefficients, we compute the large-degree asymptotics of in Dₙ(1). Finally, we derive the explicit asymptotic expansion of this probability as n →∞. The second problem is concerned with the Hankel determinant D„(t, a) and the recurrence coefficients of orthogonal polynomials from a generalized Gaussian weight e-+|-t|°(A+B.0(ェ-t)),x€(-∞,∞),whereteR,A≥0,A+B≥0and a >-1. Here θ(z) is the Heaviside step function. By using the ladder operator technique, we show that the recurrence coeffcients satisfy a particular Painlevé IV equation and the logarithmic derivative of the associated Hankel determinant satisfies the Jimbo-Miwa-Okamoto σ form of the Painlevé IV. The asymptotics of the recurrence coefficients and the Hankel determinant are obtained at the hard-edge limit and can be expressed in terms of the solutions to the Painlevé XXXIV and the o-form of the Painlevé I equation at the soft-edge limit. In addition, we consider the special case A=0,B= 1. We show that D„(t, a)can be expressed as another Hankel determinant Dₙ(t, α) associated to a semi-classical Laguerre weight ωₐ (ᵡ ; t)= ᵡᵃe⁻ˣ²⁻²ₜₓ, which is a continuous weight on [0, ∞). Then we study the recurrence coefficients of monic polynomials orthogonal with respect to the weight ωₐ (ᵡ ; t). Through a pair of ladder operators, we show the recurrence coefficients satisfy a particular Painlevé IV equation and the logarithm of the Hankel determinant Dₙ(t, α)has an integral representation in terms of the recurrence coefficients. By using the coulomb fluid approximation, we analyse the asymptotic behavior of the recurrence coefficients and the sequences of monic polynomials. After the appropriate double scaling t = √2n T, expansions of the scaled determinant Dₙ(√2n T, α) are obtained for large n and T ∈(-1, 1).