In this paper, we study the following parabolic problem of Kirchhoff type with logarithmic nonlinearity:... ut + M([u] 2s)LK u = |u| p-2u log |u|, in similar to x (0,+8), u(x, t) = 0, in (RN \ similar to) x (0,+8), u(x, 0) = u0(x), in similar to, where [u] s is the Gagliardo seminorm of u, similar to. RN is a bounded domain with Lipschitz boundary, 0 < s < 1, LK is a nonlocal integro-differential operator defined in (1.2), which generalizes the fractional Laplace operator (- similar to)s, u0 is the initial function, and M : [0,+8). [0,+8) is continuous. Let J (u0) be the initial energy (see (2.1) for the definition of J), d > 0 be the mountain-pass level given in (2.4), and similar to M. (0, d] be the constant defined in (2.6). Firstly, we get the conditions on global existence and finite time blow-up for J (u0) = d. Thenwestudy the lower and upper bounds of blow-up time to blow-up solutions under some appropriate conditions. Secondly, for J (u0) = similar to M, the growth rate of the solution is got. Moreover, we give some blow-up conditions independent of d and study the upper bound of the blow-up time. Thirdly, the behavior of the energy functional as t. T is also discussed, where T is the blow-up time. In addition, for J (u0) = d, we give some equivalent conditions for the solutions blowing up in finite time or existing globally. Finally, we consider the existence of ground state solutions and the asymptotical behavior of the general global solution.