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Motivated by the recent discovery of superconductivity in UTe$_2$, we analyze the stability and nodal structure of various triplet superconducting order parameters. Using a combination of symmetry group-theoretical analysis, phenomenological Landau free energy and weak-coupling BCS theory, we show that chiral nonunitary superconducting order can be stabilized on the border of ferromagnetism in UTe$_2$, even in the absence of long-range magnetic order. We further perform first principles density functional theory (DFT) calculations of the so-called "small" Fermi surface, excluding the contribution of U $f$-electrons, and find it to be in excellent agreement with the recent angular resolved photoemission study and DFT+DMFT calculations. This permits us to elucidate the nodal structure of the superconducting gap, which we find generically to possess point nodes along the crystallographic $a$ direction, in agreement with experiments. The topological stability of these point nodes and their associated Majorana surface states is analyzed. The nonunitary structure of the predicted superconducting state supports chiral edge modes observed in recent scanning tunneling microscopy (STM) data and is predicted to result in a non-vanishing magneto-optical Kerr effect.