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We obtain new results pertaining to convergence and recurrence of multipleergodic averages along functions from a Hardy field. Among other things, weconfirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] andobtain combinatorial applications which contain, as rather special cases,several previously known (polynomial and non-polynomial) extensions ofSzemeredi's theorem on arithmetic progressions [BL96; BLL08; FW09; Fra10;BMR17]. One of the novel features of our results, which is not present inprevious work, is that they allow for a mixture of polynomials andnon-polynomial functions. As an illustration, assume$f_i(t)=a_{i,1}t^{c_{i,1}}+\cdots+a_{i,d}t^{c_{i,d}}$ for $c_{i,j}>0$ and$a_{i,j}\in\mathbb{R}$. Then  $\bullet$ for any measure preserving system $(X,{\mathcal B},\mu,T)$ and$h_1,\dots,h_k\in L^\infty(X)$, the limit$$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T^{[f_1(n)]}h_1\cdotsT^{[f_k(n)]}h_k$$ exists in $L^2$;  $\bullet$ for any $E\subset \mathbb{N}$ with $\overline{\mathrm{d}}(E)>0$there are $a,n\in\mathbb{N}$ such that $\{a,\,a+[f_1(n)],\ldots,a+[f_k(n)]\}\subset E$.  We also show that if $f_1,\dots,f_k$ belong to a Hardy field, have polynomialgrowth, and are such that no linear combination of them is a polynomial, thenfor any measure preserving system $(X,{\mathcal B},\mu,T)$ and any$A\in{\mathcal B}$, $$\limsup_{N\to\infty}\frac{1}{N}\sum_{n=1}^N\mu\Big(A\capT^{-[ f_1(n) ]}A\cap\ldots\cap T^{-[f_k(n)]}A\Big)\,\geq\,\mu(A)^{k+1}.$$

Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications