<p>In this paper we study the total positivity of almost-Riordan arrays <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((d(t)|\, g(t), f(t))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mspace width="0.166667em" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and establish its necessary conditions and sufficient conditions, particularly, for some well-used formal power series <i>d</i>(<i>t</i>). We present a semidirect product of an almost-array and use it to transfer a total positivity problem for an almost-Riordan array to the total positivity problem for a quasi-Riordan array. We find the sequence characterization of total positivity of the almost-Riordan arrays. The production matrix <i>J</i> of an almost-Riordan array <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((d|\, g,f)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">|</mo> <mspace width="0.166667em" /> <mi>g</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is presented so that the total positivity of <i>J</i> implies that of both the almost-Riordan array <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((d|\, g,f)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">|</mo> <mspace width="0.166667em" /> <mi>g</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the Riordan array (<i>g</i>,&#xa0;<i>f</i>). We also present a counterexample to illustrate that this sufficient condition is not necessary. If the production matrix <i>J</i> is tridiagonal, then the expressions of its principal minors are given. By using these expressions, we find a sufficient and necessary condition of the total positivity of almost-Riordan arrays with tridiagonal production matrices. Numerous examples are given to demonstrate our results.</p>